024562-1-F THEORETICAL CHARACTERIZATION OF MICROSTRIP LUMPED ELEMENTS AND INTERCONNECTS IN M3IC'S by Pisti B. Katehi Dimitris Pavlidis Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122 June 1988 1

PROGRESS REPORT 1. ARO Proposal Number: DAAL03-87-K-0088 (23836-EL) 2. Period Covered By Report: 1 January 1988 - 30 June 1988 3. Title of Proposal: Theoretical and Experimental Study of Microstrip Discontinuities in Millimeter Wave Integrated Circuits. 4. Contract Number: DAAL03-87-K-0088 5. Name of Institution: University of Michigan 6. Authors of Report: Pisti B. Katehi Dimitris Pavlidis 7. Listing of Interim Reports and Manuscripts Submitted or Published Under Full or Partial ARO Sponsorship During This Reporting Period: (1) L.P. Dunleavy and P.B. Katehi, "A New Method For Discontinuity Analysis in Shielded Microstrip," Digest of the 1988 IEEE MTT-S International Symposium, New York, New York, May 1988, pp. 701704. (2) L.P. Dunleavy and P.B. Katehi, "Shielding Effects in Microstrip Discontinuities- Part I: Theory". It has been submitted for publication in the IEEE Trans. on Microwave Theory and Techniques. (3) L.P. Dunleavy and P.B. Katehi, "Shielding Effects in Microstrip Discontinuities- Part II: Applications". It has been submitted for publication in the IEEE Trans. on Microwave Theory and Techniques. 2

(4) T.E. van Deventer and P.B. Katehi, "High Frequency Conductor Losses in Shielded Microstrip". Technical Report ARO-024562-2-T, EECS Department, University of Michigan, Ann Arbor, June 1988. (5) M.E. Coluzzi and P.B. Katehi, " Theoretical Characterization of an Air-Bridge". Technical Report ARO-024562-1-T, EECS Department, University of Michigan, Ann Arbor, June 1988. 8. Scientific Personnel Supported By This Project: Faculty Graduate Students P. B. Katehi M. Coluzzi D. Pavlidis M. Weiss W. Harokopus E. van Deventer 3

RESEARCH TASKS Title Personel Involved with the Research 1.Theoretical Characterization P.B. Katehi* of an Air-Bridge. 2.High Frequency Conductor P.B. Katehi Losses in Shielded Microstip. T.E. van Deventer 3.Shielding Effects on the P.B. Katehi Phase Velocity of Microstrip L.P. Dunleavy** Lines. 4.Gate Capacitance Modeling of D. Pavlidis GaAs MESFET's in Millimeter M. Weiss Wave Frequencies. W. Harokopus P.B. Katehi * Part of the theoretical formulation for this problem has been carried out by Mr.M. Coluzzi who was a graduate student supported by this contract during the period September 1987-January 1988. Last February, Mr. Coluzzi left the University and since then Dr. Katehi has been responsible for this task. ** Dr. L.P. Dunleavy graduated last May and he has been with Hughes Torrance since then. 4

1. THEORETICAL CHARACTERIZATION OF AN AIR-BRIDGE Faculty Supervisor: Pisti B. Katehi Graduate Student Participant: None at the present time. ( There is a new graduate student starting in Fall 1988 who will continue the work in this problem). Period: 1 January 1988 - 30 June 1988. Work Performed: The theoretical formulation for the air bridge has been completed. The analytical method employed in these algorithms efficiently takes into account shielding effects and conductor losses(5)(Appendix E). Program for the Second Year: Using the completed theoretical formulation of the problem, computer programs will be written which will give frequency dependent equivalent circuits or scattering parameters for the Air-Bridge. The theoretical results will be compared to available experimental data for verification. Publications and Reports: M.E. Coluzzi and P.B. Katehi, "Theoretical Characterization of an Air-Bridge". Technical Report ARO-024562-1-T, EECS Department, University of Michigan, Ann Arbor, June 1988. 5

2. HIGH FREQUENCY CONDUCTOR LOSSES IN SHIELDED MICROSTRIP Faculty Supervisor: Pisti B. Katehi Graduate Student Participant: T. E. van Deventer Period: 1 January 1988 - 30 June 1988 Work Performed: A new analytical method has been developed to evaluate conductor losses in single or multiple shielded microstrip lines (4)(Appendix D). Numerical results for the case of a single line have been derived and we are in the process of comparing them to available data at lower frequencies. Program for the Second year: Extensive numerical data will be derived for the case of single or multiple shielded microstrip lines printed on multilayer substrates. In addition, we will try to extend the method to microstrip discontinuities. Publications and Reports: T.E. van Deventer and P.B. Katehi, "High Frequency Conductor Losses in Shielded Microstrip". Technical Report ARO-024562-2-T, EECS Department, University of Michigan, Ann Arbor, June 1988. 6

3. SHIELDING EFFECTS ON THE PHASE VELOCITY OF MICROSTRIP LINES Faculty Supervisor: P.B. Katehi Graduate Student Participant: L.P. Dunleavy Period: 1 January 1988 - 30 June 1988 Work Performed: An integral equation method has been developed for the accurate evaluation of shielding effects on the propagation properties of shielded microstrip lines. The integral equation has been derived by applying reciprocity theorem and then is solved by the method of moments (2),(3),(Appendices B,C). Numerical results have been derived and compared to experimental ones for verification of the theory. Program for the second year: For the second year we plan on extending the study of shielding effects to single or multiple microstrip lines on multilayer substrates. Publications and Reports: L.P. Dunleavy and P.B. Katehi, "A New Method for Discontinuity Analysis in Shielded Microstrip," Digest of the 1988 IEEE MTT-S International Symposium, New York, New York, May 1988, PP. 701704. L.P. Dunleavy and P.B. Katehi, "Shielding Effects in Microstrip Discontinuities- Part I: Theory". It has been submitted for publication in the IEEE Trans. on Microwave Theory and Techniques. 7

L.P. Dunleavy and P.B. Katehi, "Shielding Effects in Microstrip Discontinuities- Part II: Applications". It has been submitted for publication in the IEEE Trans. on Microwave Theory and Techniques. 8

4 SIMULATION OF TRANSISTOR INPUT IMPEDANCE AT MILLIMETRWAVE FREQUENCIES. Faculty Supervisors: D. Pavlidis P.B. Katehi Post Graduate Participant: M. Weiss Graduate Student Participant: W.P. Harokopus Work Performed: As a first step to the analytical evaluation of a GaAs MESFET at millimeter wave frequencies, a quasi-static modeling of the transistor gate was caried out using a finite difference method.In order to reduce computation time, we used Neumann's boundary conditions in conjuction with a graded mesh. Results which were derived from this approach compare very well with available data for the case of a simple-shaped depletion region. In addition, we have initiated a dynamic analysis of the problem which is based on a finite element method. The formulation of the problem has been completed and we are in the process of writing the computer programs. Program for the Second Year: In addition to the gate capacitance modelling we are planning to extend our theoretical studies towards a better understanding of other transistor characteristics such as gate-drain and drain-source capacitance, access resistance to the active channel, and geometry and transition effects. 9

TASKS TO BE INITIATED DURING THE SECOND YEAR Other tasks we plan to initiate during the coming period is a study on the geometry optimization of the devices. Efficient millimeter wave operation requires a good knowledge of the electromagnetic field transformation at the transmission line to gate or drain terminal transition. Propagation along the gate and drain strips should also be considered because of the effects that it can have on the transistor gain. The gate configuration, width and paralleling for power applications will be examined and understood by considering the distributed nature of the device. Using our full-wave analysis we will address the above problems and extract equivalent circuits permitting the optimization of device geometry and material parameters. For these tasks, Dr. D. Pavlidis and Dr. P.B. Katehi will serve as faculty advisors. Other scientific personel who will participate in these problems are: T.E. van Deventer (Graduate Student) M. Weiss ( Postdoctoral Fellow ) 10

Appendix A "A new Method for Discontinuity Analysis in Shielded Microstrip" L.P. Dunleavy and P.B. Katehi 11

BB-1 A NEW METHOD FOR DISCONTINUITY ANALYSIS IN SHIELDED MICROSTRIP L.P. Dunleavy and P.B. Katehi The Radiation Laboratory University of Michigan, Ann Arbor, MI Super Compact and Touchstone 1, and to measurements. Abstract-. A new integral equation method is described The measurements were performed using a variation of for the accurate full-wave analysis of shielded microstrip the TSD de-embedding technique [8,9]. discontinuities. The integral equation is derived by an application of reciprocity theorem, then solved by the method of moments. Numerical and experimental results are presented for open-end and series gap disconti- microstrip shielding nuities, and a coupled line filter. cavity (or housing), \ Y coaxial, coaxial I. INTRODUCTION int output The development of more accurate microstrip discon-. O — y= tinuity models, based on full-wave analyses, is key to zh. ^ improving microwave and millimeter-wave circuit simulations and reducing lengthy design cycle costs. In most x X=a applications, radiation and electromagnetic interference / are avoided by enclosing microstrip circuitry in a shield- dielectric substrate ing cavity (or housing) as shown in Figure 1. The effect of the shielding is significant, and requires accurate mod- Figure 1: In most practical designs, microstrip circuitry is enclosed eling, at high frequencies. Shielding effects are not ad- in a shielding cavity whose effects must be accurately modeled at equately accounted for in the discontinuity models used high frequencies. in most available microwave CAD software. To address these inadequacies, a new method was developed for the full-wave analysis of discontinuities in II. SUMMARY OF THEORETICAL METHOD shielded microstrip [2]. This method accurately takes into account the effect of the shielding enclosure. The In the theoretical derivation [2], an application of reciinto account the effect of the shielding enclosure. The theoretical contribution, as compared to previous work procity theorem results in an integral equation relating [3]-[5], is in the novel way that reciprocity theorem, the the magnetic current source Mo, and the electric curis i the novel way that reciprocity theorem.t rent on the conducting strips J., to the electromagnetic method of moments, and transmission line theory are rent on the conducting strips to the electromagnetic method of moments, and transmission line theory are fields inside the cavity. A Galerkin's implementation of combined to solve for discontinuity parasitics. As illus- fields de the cavitys A Galerkin's implementation of the method of moments is employed by first dividing the trated in Figure 2, the coaxial feed is modeled using an th e method of moments is em ed b first iiin then exp equivalent magnetic "frill" current [6,7]. To the authors' strips nto subsections. The current then expanded knowledge, this is the first time that the frill current ap- according to [ proach has been applied to microstrip circuit problems. To demonstrate the method, numerical results are presented for open-end and series gap discontinuities, N. and a four resonator coupled line filter. These results J = (Y) E Ipap (x) x (1) are compared to other full-wave analyses, to data from p=l 1 Swper Compact and Tochsctone are microwave CAD software packages available from Compact Software and EESOF respectively. 701 0149-645X/88/0000-070 1$01.00 ~ 1988 IEEE 1988 IEEE MTT-S Digest

fc, which is defined as the lowest frequency where nonmagnetic frill evanescent waveguide modes can exist within the cavity. coaxial current M The new results are almost identical to those obtained feed by Jansen et. al. for frequencies above 8 GHz, but show a reduced value for lower frequencies. 0.45 0.40 |, 7.9 GHZ annular aperture 035 - JANSEN 0.s THiS ESERCH -J -"" frOH Figure 2: The coaxial feed is represented by an equivalent mag- 0.25 netic frill current M, = MSS; this is used as the excitation mechanism for computing the microstrip current. 0.20 0.15. -. 0 4 8 12 16 20 24 28 FREQUENCY (HZ) where +(y) describes the variation of the longitudinal current in the transverse (i.e. y) direction, and cap(x) are sinusoidal subsectional basis functions. Figure 3: Effective length extension of a microstrip open-end disThe resulting equation may be expressed as continuity, as compared to results from other full wave analyses ((r = 9.6, W/h = 1.57, b =.305",c =.2", h =.025"). EZIIj Es,(x = h) * (y) ap (x) dS, Is = p=l = The results shown in Figure 4 illustrate that shielding / Hq * M.ds (2) effects are significant at high frequencies. The normalJ t Jized open-end capacitance cp is plotted f'(, three difwhere S, is the surface area of the pth subsection, S1 is ferent cavity sizes. The results show that; ducing the the surface of the coaxial aperture, and Eq, H, are the cavity size raises fc (as expected), and it lowi!-rs the value electric and magnetic fields respectively, associated with of c,. For comparison, data obtained froi uper Coma test current J Jexisting over the qth strip subsection. pact and Touchson are included. We may express (2) by the matrix equation [Z] [I] = [V]. (3) 3.5 Here, [Z] is the impedance matrix, [V] is the excitation 3.0 vector and [I] is the unknown current vector comprised of the complex coefficients Ip. - 25 ERC20.1GHZOMP Finally, after evaluating the elements of [Z] and [V], 2.0 \ T LECOCST the matrix equation is solved to compute the current ig 2 A ^ CA^Ycc CAVTTY CC distribution. Based on the current, transmission line 1.5 CAVIYYCF theory is used to derive scattering parameters, and (if fc.43.GHZ desired) an equivalent circuit model, to characterize the f. 37.5GHZ discontinuity [1,2]. 0.s..5.. 0 1 20 2 30 40 50 60 70 FREQUENCY (GHZ) m. RESULTS Figure 4: A comparison of the normalized open-eudl capacitance An open-end can be represented by an effective length for three different cavity sizes shows that shielding tefects are significant at high frequencies ((, = 9.7, W = h =.025"; cavity CA: extension L.ff, by a shunt capacitance cp, or by the b = c =.25", cavity CC: b = c =.01", cavity CF: b = c =.075"). associated reflection coefficient rp (= Sii). The plot of Figure 3 compares Le// results to those of Jansen et. al. [3] and Itoh [4]. Also shown is the cut-off frequency 702

In the remaining examples, numerical results from the a, b new method are compared to measurements. Figure 5. shows results for the angle of Sin of an open-end, and Figure 6 contains results for the magnitude of the trans- mission coefficient (/S23/) for a series gap discontinuity. In both cases, the agreement between the numerical and I s experimental data is very good. t 1s w - 11.9 -lI1E I i -^7HfJ ISRESEARC! i S^ TOUCHSTONaE b'.2D.S, M MEASUREMENT L3 *C _ 0 -53 25- ARE IN MILS 1 82- 25.8 L2- 113 -30 83 27.2 L - 93.9 0 4 a 12 16 20 24 3 FREQUENCY(GH) Figure 7: Numerical and experimental results are compared below for this 4 resonator filter ( Er = 9.7, h =.025", b =.4", c =.25"). Figure 5: Numerical and measured results show good agreement for the angle of S11 of an open circuit (e, = 9.7, W = h =.025", b = c =.25")..5 -.20. Im I -1'0.40 -15-.>^^^ -*60 N MEASUREMENT 10' THISFESEARCH -20. SUPER COMPACT f *3.90aH -o TCA| SNE *.80 1 -, c N MEASUREMENT -25 -100 y-304 i/ 1 2 4 6 8 10 12 14 16 18 20 FREOUENCY(GHZ) -35 0 4 8 12 16 20 24 a. Amplitude of S21 FREQUENCY(GHZ) 180 Figure 6: Good agreement with measurements has also been ob- 135 tained for series gap discontinuities. Shown here is the magnitude 90 of S21 for a series gap with a 9 mil gap spacing ( rc = 9.7,W = h =.025", b = c =.25"). 45 "' — THISRESEARCH -45 48 * " MEAS.EMEI/ Finally, consider the four resonator filter of Figure 7.90 Numerical results for the magnitude and phase of S21, -135 shown in Figure 8, demonstrate excellent agreement with -18.,.,..,-".measurements for frequencies below the cutoff frequency 2 4 10 12 14 16 1 20 fc. Above cutoff, the filter measurement is distorted due FREmucENCY(H to waveguide moding within the test fixture. b. Phase of S2i Figure 8: Results for transmission coefficient S21 of 4 resonator filter. 703

[7] R. Harrington, Time-Harmonic Electromagnetic IV. CONCLUSIONS Fields, McGraw Hill 1961, pp.111-112. A new analysis method has been described for shielded [8] N. Franzen and R. Speciale, "A New Procedure for microstrip discontinuities. Results from this method System Calibration and Error Removal in Autohave demonstrated good agreement with measurements mated S-Parameter Measurements", 5th European and other numerical results. This method is useful for Microwave Conference, pp. 69-73. the evaluation of existing discontinuity models, for the analysis of cases where existing solutions fail -such as [9] L. Dunleavy and P. Katehi, "Repeatability Iswhen shielding effects are significant-, and for the de- sues for De-embedding Microstrip Discontinuity Svelopment of new discontinuity models with improved parameter Measurements By the TSD Technique" accuracy for high frequency applications. Automatic RF Techniques Group (ARFTG) Conf. Dig. June 1986. ACKNOWLEDGEMENTS The authors thank Mr. Ed Watkins, Mr. Jim Schellenberg, and Mr. Marcel Tutt for their contributions to this work. This work was primarily sponsored by The National Science Foundation (Contract. No. ECS -8602530). Partial sponsorship was also provided by the Army Research Office (Contract No. DAAL03-87-K0088) and the Microwave Products Division of Hughes Aircraft Co. REFERENCES [1] P. Katehi and N. Alexopoulos,"Frequency Dependent Characteristics of Microstrip Discontinuities in Millimeter-wave Integrated Circuits", IEEE Trans. Microwave Theory Tech. Vol. MTT-33 No. 10, Oct. 1985, pp. 1029-1035. [2] L. Dunleavy and P. Katehi, "A New Method for Discontinuity Analysis in Shielded Microstrip: Theory and Computational Considerations", In preparation. [3] R. Jansen, and N. Koster, "Accurate Results on the End Effect of Single and Coupled Lines for Use in Microwave Circuit Design " A.E. U. Band 34 1980, pp 453-459. [4] T. Itoh,"Analysis of Microstrip Resonators", IEEE Trans. Microwave Theory Tech., Vol Mtt-24 1974, pp 946-951. [5] J. Rautio, "An Electromagnetic Time-Harmonic Analysis of Shielded Microstrip Circuits",IEEE Trans. Microwave Theory Tech., Vol. MTT-35, No. 8, pp726-729. [6] C. Chi and N. Alexopoulos, "Radiation by a Probe Through a Substrate" IEEE Trans. Antennas Propagat. vol. AP-34, Sept. 1986, pp 1080-1091. 704

Apmendix B "Shielding Effects in Microstrip Discontinuities Part I: Theory" L.P. Dunleavy and P.B. Katehi 12

Shielding Effects in Microstrip Discontinuities - Part I: Theory Submitted to IEEE Trans. on Microwave Theory and Tech. - April 1988 L.P. Dunleavy* and P.B. Katehi Radiation Laboratory Dept. of Electrical Engineering and Computer Science The University of Michigan 1301 Beal Avenue Ann Arbor, MI 48109-2122 Abstract-. A new integral equation method is described for the accurate full-wave analysis of shielded microstrip discontinuities. The integral equation is derived by applying the reciprocity theorem, then solved by the method of moments. In this derivation, a coaxial aperture is modeled with an equivalent magnetic current, and is used as the excitation mechanism for generating the microstrip currents. Computational aspects of the method have been explored extensively. A summary of some of the more interesting conclusions is included. L.P. Duneavy is now with Huoes Aircrft Company.

I. INTRODUCTION The need for more accurate microstrip circuit simulations has become increasingly apparent with the advent of monolithic microwave integrated circuits (MMICs), as well as the increased interest in millimeter-wave and near-millimeter-wave frequencies. The development of more accurate microstrip discontinuity models, based on full-wave analyses, is key to improving high frequency circuit simulations and reducing lengthy design cycle costs. Further, in most applications the microstrip circuit is enclosed in a shielding cavity (or housing) as shown in Figure 1. There are two main conditions where shielding effects are significant. The first occurs when the frequency approaches or in above the cutoff frequency fc for higher order modes. The second occurs when the metal enclosure is physically close to the circuitry. A full-wave analysis is required to accurately model these effects. Although shielding effects have been studied to some extent in the past (e.g. [1]), the treatment has been incomplete, particularly for more complicated structures such as a coupled line filter. Further, shielding effects are not accurately accounted for in the discontinuity models of most available microwave CAD software. To address these inadequacies, Part I of this paper develops an accurate method for the analysis of discontinuities in shielded microstrip. The method presented is based on an integral equation approach. The integral equation is derived by an application of reciprocity theorem, then solved by the method of moments. To derive a realistically based formulation, a coaxial excitation mechanism is used. To date, all full-wave analyses of microstrip discontinuities use either a gap 1

generator excitation method [2,3,6], or a cavity resonance technique [4,5]. Both of these techniques are purely mathematical tools. The former has no physical basis relative to an actual circuit. The latter is also abstract, since in any practical circuit some form of excitation is present. In fact, one of the most common excitations in practice comes from a coaxial feed (Figure 1). A magnetic current model for such a feed is used in the present treatment as the excitation, and this method is compared to the gap generator method in Part II of the paper. The emphasis in Part II is on the application of the present theoretical method to study shielding effects in discontinuities. As part of this study, numerical and measured results are presented for the structures of Figure 2, which include openend and series gap discontinuities, and a coupled line filter. The measured results are seen to be in excellent agreement with the theoretical results from the present research. The objectives of this research may be summarized as follows: i) Develop a new theoretical method for analysis of shielded microstrip discontinuities (Part I), ii) Explore the use of a practical (i.e. coaxial) excitation mechanism (Part I and II), iii) Investigate high frequency microstrip measurement techniques (Part II), v) Study the effect of shielding on discontinuity behavior (Part II), and iv) Obtain experimental data for verification of the new theoretical method (Part II). In addition, computational aspects of the method are explored extensively. 2

II. THEORETICAL FORMULATION The details of the theoretical derivation for the present method are given in [7]. Hence, only a summary of the key steps is described below. A. Integral Equation In the theoretical formulation, a few simplifying assumptions are made to reduce unnecessary complexity and excessive computer time. Throughout the analysis, it is assumed that the width of the conducting strips is small compared to the microstrip wavelength AX (the "thin-strip" approximation). In this case, the transverse component of the current may be neglected. While substrate losses are accounted for, it is assumed that the strip conductors and the walls of the shielding box are lossless, and that the strip has infinitesimal thickness. These assumptions are valid for the high frequency analysis of the microstrip structures of Figure 2, provided good conductors are used in the metalized areas. Consider the geometry of Figure 1. In most cases the coaxial feed, or "launcher", is designed to allow only transverse electromagnetic (TEM) propagation, and the feed's center conductor is small compared to a wavelength (kr, < 1). In these cases, the radial electric field will be dominant on the aperture and we can replace the feed by an equivalent magnetic surface current M, [8]. This current is sometimes called a "frill" current [9]. The source M, induces the current distribution J. on the conducting strip and produces the total electric field Etot and the total magnetic field HR inside the cavity as indicated in Figure 1. Now consider a cavity geometry similar to Figure 1, with the strip conductors as well as the coaxial input and output removed. Assume a test current Jq existing 3

on a small subsection of the area which was occupied by the strip. The fields inside this new geometry are denoted by E,, and Hq. Using the reciprocity theorem, the two sets of sources (Mo, Jo; and Jq) are related according to // J (J.s E-H, M. )dv =//, J.Jotd (1) where V represents the volume of the interior of the cavity. Since Jq * Ett is zero everywhere inside the cavity, the right hand side of (1) vanishes. Reducing the remaining volume integrals in ( 1) to surface integrals results in f/ Eq(z = h) * Jds = J Hq(x = 0) * Mds (2) where S,,tip is the surface of the conducting strip and Sf is the surface of the coaxial aperture(s). For one-port discontinuities, Sf represents the surface of the feed on the left hand side of Figure 1, while for two-port discontinuities, Sf represents both feed surfaces. An integral equation similar to (2) can be derived for the case of gap generator excitation by setting M, = 0 and assuming that E. is non-zero at one point on the strip [7]. In order to solve the integral equation (2), the current distribution J. is expanded into a series of orthonormal functions as follows 1: N. J. = (y) Ipa, (x)X (3) p=1 where Ip are unknown current coefficients and N. is the number of sections considered on the strip (Figure 3). The function (y) describes the variation of the 1Thime dependence is e'. 4

current in the transverse direction and is given by [2,10] Yo-W/2 < y Yo+W/2 (y)= W- /-[ ](4) 0 otherwise where W is the width of the microstrip line and Yo is the y-coordinate of the center of the strip with respect to the origin in Figure 1. The basis functions ap(x) are described by in[K(x+l-)l: p+l (XV+1-4 x < x < x,+x (: ( in(K) = < Kxp} < Xp (5) =. ain(Kt) np-1 - - 0 otherwise for p 6 1, and sin[K(fl-x)1 0 < < a, (x) = in(Kl) (6); 0 otherwise for p = 1, where K is a scaling factor, taken to be equal to the wave number in the dielectric x, is the x-coordinate of the pth subsection (= (p- 1)/) I, is the subsection length (l, = p+l - p). For computation, all of the geometrical parameters are normalized with respect to the dielectric wavelength (Ad); hence the normalized scaling factor is equal to 27r. The integral equation (2) can now be transformed into a matrix equation by substituting the expansion of (3) for the current J. The result may be put in the form [z[i = V]. (7) 5

In the above, [Z] is an N. x N. impedance matrix, [I] is a vector comprised of the unknown current coefficients Ip, and [V] is the excitation vector. The individual elements of the impedance matrix are given by ZQP = E(z = h) * X^ (y) ap (x) ds-. (8) where Sp is the area of the two subsections on either side of the point zp. The elements of the excitation vector are found according to Vq= jHs Mgds. (9) Once the elements of the impedance matrix and excitation vector have been computed, the current distribution is found by solving (7) as follows: [I] = [l- V]. (10) B. Evaluation of impedance matrix elements Before evaluating the elements of the impedance matrix, the Green's function associated with the electric current Jq is derived. To do this the cavity is divided into two regions: region 1 consists of the volume contained within the substrate (z < h), while region 2 is the volume above the substrate surface (z > h). The integral form of the electric field is given in terms of the Green's function, by Illv-^y [(J k2 ) ] (11) where kj = w2lsof. The index i indicates that the above holds in each region (i.e. for i = 1,2). In (11), Gs is a dyadic Green's function [11] satisfying the following equation V2 G + kG - = 16 (r-f). (12) 6

where I is the unit dyadic (= xx + yy + ii), I is the position vector of a field point anywhere inside the cavity, and r' is the position vector of an infinitesimal current source. Because of the existence of an air/dielectric interface, and the assumption of a unidirectional current, the dyadic Green's function will have the form G'= G'x + Ixz. (13) The dyadic components of (13) are found by applying appropriate boundary conditions at the walls: x = 0, and a; y = 0, and b; and z = 0, c; and at the air-dielectric interface [7]. These components may be expressed as 00 00 G(1)- A A ) cos kz sin ky sin k1)z (14) m=1 n=O G(1) - E BB() sin kx sin ky cos k?)z (15) m=l n=O G = E 2 A() cosk.z sinkyy ink2)(z- c) (16) m=l n=O G(2) =,EB() sin k. sin k,y cosk(2)(z-2 c) (17) m=l n-= where k. = nr/a (18) k, = mmr/b (19) ) = kV?- k-k2 (20) k )= k - k - k (21) ki = w/jiio (22) ko = WtIoe, (23) 7

and A -) -yn cos k 8x' sin k y' tan k()(h - c) ) abdlmn cos 1')h A2) = -Yn cos k_,x' sin kyy' tan k)h (25 "mn - ~ i(2)/. \''(25) abdlmn cos k(h - c) B(1) = -Yn(l -;) k cos kz' sin kvy' tan k()h tan k(2)(h-c) (26) abdlmnd2mn cos k )h B(2) _ - -n(l - e*)ks cos kz' sin k y' tan k()h tan k(2)(h - c) -mn (27) abdimnd2mn cos k 2(h - c) In (24)-(27), e* is the complex dielectric constant of the substrate and 2 for n = 0 Yn (28) 4 for n 0 dimn = k2) tan k(l)h - k') tan k2)(h - c) (29) d2mn = kc2) tank(2)(h - c) - k) tan kc)h. (30) In view of (11)-(30), the elements of the impedance matrix may be put in the following form 2: jwpnoK212 NSTOP zqp = ab sin2 Kl p n cos kxq cos kxp.[Sinc [(k. + K)1J] Sinc [1(k - K)l]]2 LN(n) (31) with LN(n) given by the series MSTOP LN(n)= E Lmn. (32) m=l The series elements L,n are given by 2The expresioa given here for the impace mtrix elements, and that given shortly for the excitation vector elements apply to the cse of an open-end or series gap. it modificatios are necesry for analyis of paralel coupled line filter. 8

L np[sin(kYo)Jo (k-)]2 tan k )h tan k,2)(h - c) m ~ [k(2) tan k(1)h - k() tan )(h- c)] [k2)e (i - ) tan k(2)(h - c) - k() (1 - tan k(I)h] [kXe tan k()(h - c) - tan k(.)h] where Yo is the y-coordinate of of the center of the strip, and {"^ for t 0 Sinc(t) = (34) 1 fort =0 C = 2 for q= 1 Cq = (35) 4 otherwise RIn = X(k +kx)ls (36) R2n =.(k-ks)l. (37) C. Evaluation of the Excitation Vector Elements The formulation for the excitation vector elements for the one-port case will now be carried out. The case for two-port excitation is a straightforward extension [7]. To evaluate the excitation vector elements according to (9), we need to find the magnetic field Hq and the frill current M. = M,*. An approximate expression for the frill current is given by [8] iM~. = - o- (38) where V0 is the complex voltage applied by the coaxial line at the feeding point rb is the radius of the coaxial feed's outer conductor 9

r, is the radius of the coaxial feed's inner conductor p, a are cylindrical coordinates referenced to the feed's center. Substituting from (38) into (9) yields (with ds = pdpdo) - V0 VI, = I| HO(x = O)dpdOq V0 jfI [/JH()=(z=O)dpd + f H(2) (= O)ddl] (39) where S(1) is the portion of the feed surface below the substrate/air interface (z" = psin < -t) S(2) is the portion of the feed surface above the substrate (z" = p sin f >- -t) H(')(x = 0) and H()(z = 0) are the f components of the magnetic field, in regions 1 and 2 respectively, evaluated on the plane of the aperture. After solving for the magnetic fields Hi(xz = 0) and substituting the resulting expressions into (39), the following formulation is produced for excitation vector elements: = -V4Kl 2 NSTOP Vq - cosVokqKl_ ln(,) 4absinKl.,Eo Sinc [(ks + K)4l] Sinc [2(k. -K)l-] [MN(n)] (40) where MN(n) is expressed in terms of the series given by MSTOP MN(n)= A Mmn. (41) mThe series elements M are given by the following integral The series elements Mmn are given by the following integral 10

Mm n Mn J M n dp d Mm = JJ M(1) dpdo+f M(2) dp d. (42) The above integrations are performed numerically, with the integrands MMni given by M() =c cos ) cos k,(pcos + Y) sin k()(p sin + h,) -sin c(,) sin ky(p cos q+ Yc) cos k()(p sin d + hc) (43) for p and' in region 1, and M(2) _= C oC cos ky,(p os 0s + YC) sin k(2)(p sin -c + h) -sin c(2) sin ky(p cos + Yc) cos k)(p sin - c + h). (44) for p and 0 in region 2. In (43), and (44) Y,, and h, are the the y and z coordinates of the coaxial feed, and - [(k1))2 + k2(1 - )] tan k()h} (45) -(i) ynky tanlk(2)(h - c) Wk ""mn = dimk coYosJo(kT) (46) c() = tan nk(h, - ) sin kv Yo Jo(k, ) (46) cmn =.'~ {k)2 tan kl)h - [(k())2r_ - k(1 -e)] tan k2)(h-c)} (47) - 2) pk., tank )h. (48) C(2n Wn ~ 2 _ sin - sYo Jo(kV (48) dimncos k)(h - c) 2 The above outlines the theory for computing the current distribution on the conducting strips of shielded microstrip discontinuities. The next step is to use the 11

current distribution to derive the network parameters of the discontinuity under consideration. As shown in Figure 4, an open-end discontinuity can be represented by an effective length extension Ljff or by an equivalent capacitance Cop. For a two-port network, a general scattering parameter representation is used. The effective dielectric constant Eeff is calculated from the distance between two adjacent maxima of the current distribution on a straight microstrip line. The theory for deriving these network parameters is described for elsewhere [2,7,12], and a brief summary is given in Appendix I. The theoretical method developed above has been implemented in.a Fortran program. The remainder of the paper addresses computational aspects of the solution for the current distribution and discontinuity network parameters. III. COMPUTATION OF CURRENT DISTRIBUTION To gain insight into the nature of the computations, we will now examine plots of a typical impedance matrix, excitation vector, and current distribution for an open-ended microstrip line. Figure 5 shows the amplitude distribution of a typical impedance matrix. It is seen that the amplitude of the diagonal elements is the greatest and it tapers off uniformly as one moves away from the diagonal. Another observation is that the matrix is symmetric such that Zqp = Zpq for any p and q, which is expected from (31). When the impedance matrix of Figure 5 is inverted, the amplitude distribution is as shown in Figure 6. The inverted impedance matrix shows a sinusoidal shape for any given row or column. Figure 7 shows the amplitude distribution for the excitation vector. The ampli12

tude is highest over the subsection closest to the feed then tapers off smoothly. In contrast, the excitation vector for the gap generator method has only one non-zero value, at the position of the source. Multiplying the inverted impedance matrix by the excitation vector of Figure 7 yields the current distribution of Figure 8. It can be seen that the shape of the current is similar to that exhibited by the first column of the impedance matrix. This is not surprising given the shape of the excitation vector. IV. CONVERGENCE OF Zqp AND Vq In the expressions of (31), and (40) for the impedance matrix and excitation vector elements, the summations over m and n are theoretically infinite. The number of elements included in these series depends on the convergence behavior of Zqp and Vq with the summation indices. As seen from (31), the convergence of the impedance matrix is described mainly by the convergence of LN(n). Figure 9 shows the typical variation of LN(n) with m and n. Most of the contributions from LN(n) to the impedance matrix are concentrated in the first several n values. The convergence over m is good, and it appears that performing the computations out to m = 200 may be sufficient. Note, however, that the allowable truncation points for the summations over m and n vary with the geometry. The values quoted here are for illustration purposes only. The computation of Zqp over n is illustrated for a typical impedance matrix in Figure 10. Shown is the convergence behavior for one row (q = 32) of the 64 x 64 element impedance matrix of Figure 5. This behavior is representative of that for 13

any row. After only a few terms the diagonal element (p = q = 32) rises above the others, and after adding 100 terms the amplitude distribution is well formed. Similar conclusions can be drawn for the convergence of the excitation vector elements with respect to the summation indices m and n. V. CONVERGENCE OF NETWORK PARAMETERS The convergence behavior of the elements of the impedance matrix and excitation vector is important to examine, yet the more relevant question remains: how are the final results affected by various convergence related parameters? To answer this question, a series of numerical experiments were carried out, and the main results are presented here. As illustrated in Figure 4, an open end discontinuity can be represented by either an effective length extension Leff or an equivalent capacitance cp. The microstrip effective dielectric constant Eeff is calculated from the distance between two adjacent maxima of the open-end current distribution (Figure 8). The experiments investigated the convergence behavior of Leff and e!ff with respect to the sampling rate Ns (= 1/1), and the truncation points NSTOP, MSTOP for the summations over n and m respectively. These numerical experiments have been grouped into three separate categories each exploring a different aspect of the convergence behavior 3. 3The paramete used for the plots shown in this ectimn are the following:, = 9.7, W = h =.025", = 3.5", b = c =.25", f = 18GHs. 14

A. Effect of K-value Using the program mentioned above, data was generated to plot Leff and seff versus N. for several different values of the normalized scaling factor K of (5) and (6). Figure 11a shows the convergence behavior of Leff for a typical case. It is seen that a relatively flat convergence region exists for all the K-values between about 40 and 100 samples per wavelength. Outside this region the convergence behavior depends on K. At first glance, it appears that the best convergence is achieved for higher K-values (e.g. K = 8ir); however, quite the opposite conclusion results from examining the eeff computation. As can be seen from Figure 1lb, the best convergence for esof is obtained for low K-values. Based on these and other observations [7], it was determined that a value of K = 2r gives the best convergence behavior for the Leff and eeff computations. B. Leff, Eeff Convergence on n and m To investigate the convergence of the network parameter computations with the summation index n, several program runs were executed for different values of NSTOP, with MSTOP fixed at 1000. Data was generated to plot Lff and ecff versus n for several 1, values. Figure 12a shows that for all the l1 values, good convergence on n is achieved after 500 terms. The same can be said for the convergence of eel.f In examining the convergence behavior with n it was found that, for a given subsection length 1X, cavity length a, and truncation point NSTOP, a maximum sampling limit exists beyond which the computed current becomes completely 15

erratic. This is called the erratic current condition and is given by the following simple relationship: NSTOP NSTOP < a or N > N- -. (49) a To investigate the convergence behavior with respect to the summation index m, NSTOP was fixed at 500, and the program was run for different values of MSTOP. Figure 12b shows that Leff converges well on m after about 500 terms. The convergence behavior of ceff on m, was found to be similar to that for Leff. C. Optimum Sampling Range In this last numerical experiment, the effect of varying 14 on the numerical accuracy of the matrix solution was examined. This was done by studying the variation of the matrix condition number [13], with respect to I. for a fixed matrix size. After studying several cases it was found that an optimum sampling range, may be defined by the following choice of subsection length 1, 1.5a 4a NSTOP - NSTOP (50) Sampling within this range automatically avoids the erratic current condition and provides the best accuracy in the matrix solution, and also in the solution for network parameters. To support this last claim, consider the plot of Figure 13. It is seen that the optimum sampling region specified by (50) coincides directly with the flat convergence region for the L.ef calculation. This consistency between the optimum sampling region and the flat convergence region for the Leff calculation was observed in all the cases examined [7]. 16

VI. SUMMARY AND CONCLUSIONS In the theoretical part of the presented research, a method of moments formulation for the shielded microstrip problem was derived based on a more realistic excitation model than used with previous techniques. The formulation follows from the reciprocity theorem, with the use of a frill current model for the coaxial feed. Computational considerations for implementing the theoretical solution were studied extensively. Several numerical experiments were presented that explored the convergence and the stability of the solution. Most significantly, it was found that an erratic current condition and an optimum sampling range exist; both of these are given by very simple relationships. Using the method presented here, Part II concentrates on the theoretical and experimental characterization of the discontinuities of Figure 2, and studies the effects of shielding on their behavior. 17

APPENDIX I A. One-Port Network Parameters (Open-End Discontinuity) The effective length extension (Figure 4) for an open-end discontinuity is given by Leff = - dma. (51) where dmas is the distance from the end of the line to a current maximum. The normalized equivalent capacitance (Figure 4) can be expressed as = sin 2/3gd,ax sin 23g, Le,f cOP = (1 - cos 2/gd,, ) w(1 + cos 2gL.f) (52) In the above, 3, is the phase constant of microstrip transmission line. B. Two-Port Network Parameters (Gap discontinuity, Coupled Line Filters) For the computation of two-port network parameters, the strip geometry is assumed to be physically symmetric with respect to the center of the cavity (in both the x and y directions of Figure 1). The network parameters are determined by analyzing the currents from the even and odd mode excitations as discussed in [2,7,12]. The normalized impedance parameters are given by according to ZIN + ZIN (53 Zll ~ (53) ZIN - ZI(54) where ZaN and z~N are the input impedances of the even and odd mode networks. The scattering parameters for the network may be derived using the following 18

Appendix C "Shielding Effects in Microstrip Discontinuities Part II: Applications" L.P. Dunleavy and P.B. Katehi 13

Shielding Effects in Microstrip Discontinuities - Part II: Applications Submitted to IEEE Trans. on Microwave Theory and Tech. - April 1988 L.P. Dunleavy* and P.B. Katehi Radiation Laboratory Dept. of Electrical Engineering and Computer Science The University of Michigan 1301 Beal Avenue Ann Arbor, MI 48109-2122 Abstract-. As an application of the theoretical method described in a companion paper, numerical and measured results are presented for open-end and series gap discontinuities, and a coupled line filter. Comparisons are also made to commercially available CAD package predictions. The results verify the accuracy of the new theoretical method and demonstrate the effects of shielding on discontinuity behavior. The experimental techniques used, which involve the thru-short-delay de-embedding approach, are also explained. L.P. Dunleavy is now with Hughes Aircraft Company.

relations: 2 - 1 - (55) S1l = S2z = Zl- - (56) S12 = S21 = D (56) where D = zl + 22j- z~1 (57) ACKNOWLEDGEMENTS The authors thank Mr. Ed Watkins, Mr. Jim Schellenberg, and Mr. Marcel Tutt for their contributions to this work. This work was primarily sponsored by The National Science Foundation (Contract. No. ECS -8602530). Partial sponsorship was also provided by the Army Research Office (Contract No. DAAL0387-K-0088) and the Microwave Products Division of Hughes Aircraft Co. References [1] R.H. Jansen, and N.H.L. Koster, "Accurate Results on the End Effect of Single and Coupled Lines for Use in Microwave Circuit Design " A.E.U. vol. 34, pp 453-459, 1980. [2] P.B. Katehi and N.G, Alexopoulos,"Frequency-Dependent Characteristics of Microstrip Discontinuities in Millimeter-wave Integrated Circuits", IEEE 19

Trans. Microwave Theory Tech. vol. MTT-33 No. 10, pp. 1029-1035, Oct. 1985. [3] R.H. Jansen, and W. Wertgen, "Modular Source-Type 3D Analysis of Scattering Parameters for General Discontinuities, Components and Coupling Effects in (M)MICs", Proc. 17th Eur. Microwave Conf.(Rome) 1987, pp. 427-432. [4] R.H. Jansen, "Hybrid Mode Analysis of End Effects of Planar Microwave and Millimeter-Wave Transmission Lines", Proc. Inst. Elec. Eng., vol 128, pp. 77-86, Apr. 1981. [5] T. Itoh,"Analysis of Microstrip Resonators", IEEE Trans. Microwave Theory Tech., vol MTT-22, pp 946-951, 1974. [6] J.C. Rautio, "An Electromagnetic Time-Harmonic Analysis of Shielded Microstrip Circuits",IEEE Trans. Microwave Theory Tech., vol. MTT-35, pp726729, 1987. [7] L.P. Dunleavy, "Discontinuity Characterization in Shielded Microstrip: A Theoretical and Experimental Study", Ph.D. dissertation, University of Michigan, April 1988. [8] R.F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw Hill 1961, pp.111-112. [9] C. Chi and N.G. Alexopoulos, "Radiation by a Probe Through a Substrate" IEEE Trans. Antennas Propagat. vol. AP-34, Sept. 1986, pp 1080-1091. 20

[10] J.C. Maxwell, A Treatise on Electricity and Magnetism 3rd. ed., vol. 1,New York: Dover 1954 pp 296-297. [11] C.T. Tai, Dyadic Green's Functions in Electromagnetic Theory, Intext Educational Publishers 1971. [12] P.B. Katehi, "Radiation Losses in MM-Wave Open Microstrip Filters," Electromagnetics 7, ppl37-152, 1987. [13] G.H. Golub and C.F. Van Loan, Matrix Computations, John Hopkins University Press, 1983, pp26-27. 21

microstrip shielding cavity (or housing) coaxial coaxial input output Z=C _ ~ E tot ^ Fi^Hozt y=b z=h -iiiiiiii: / Ix=a dielectric substrate Figure 1: In most practical designs, microstrip circuitry is enclosed in a shielding cavity whose effects must be accurately modeled. 22

W w OPEN END SERIES GAP k\\S\\\\ PARALLEL-COUPLED LINE FILTER Figure 2: In the present research, the theoretical method is applied to the class of discontinuity structures shown here. 23

y b u% \ \ d\ ~ % I y I %??? % - Y i I I | I I I X Ix x,... X..., X X X 1 2 3 P No-2 Ns-1 Ns X x - (p-l)l. P 1 Figure 3: The current in the longitudinal direction is expanded using overlapping sinusoidal basis functions. 24

A ffkF "' — C O OR'.4Jop \ P r r op op Figure 4: A shielded microstrip open-end can be represented by an effective length extension Le//, or by an equivalent capacitance Cop. 25

/Zqp / p Figure 5: The impedance matrix for an open-end is diagonally dominant. 26

/INVZqp / q Q=64 Figure 6: The inverted P=64 Figure 27: The inverted impedance matrix displays a sinusoidal hape along any ow or colu

0.1 83E+00 0. 146E+00 0.1 10E+00 0.730E-01 0.365E-01 -0.647E-05 0.000 0.315 0.630 0.945 1.260 1.575 X(WAVELENGTHS) Figure 7: The amplitude distribution of the excitation vector is highest at the position of the feed (z = 0), then tapers off uniformly. 28

0.560E-02 0.336E-02 0.1 1 1 E-02 -0.113E-02 -0.338E-02 -0.563E-02- 0.000 0.320 0.640 0.960 1.280 1.600 X(WAVELENGTHS) Figure 8: The imaginary part of the current distribution for an open-ended line has a sinusoidal shape, and goes to zero at the line's end (z = 1.6Ad). 29

/LN(n)/ dependenc m=200 n-dependence n=100 Figure 9: LN(n) has convergent behavior over both m and n. 30

/Zqp / P=1 N= 100 Figure 10: A row (q = 32) of the impedance matrix is seen to be well formed after adding 100 terms on n. 31

0.45 0.40 -' — K.Pl/10 e |0.35 - K.PI'-. — K.2'PI 0.3250 - *I — K.10*P1 0.20 0.15 -....... 0 25 50 75 100 125 150 Nx (samplea/wavelngth) a. A plot of Lj versus sampling displays a flat convergence region for all values of K considered. 8 8 —-- I- |~ —--- K. PI/0 CO -1 - K.-PI -.- K-PI 4 0 SO 100 150 NX (-mpeeAwv ieth) b.The convergence of er j versus sampling is seen to be better for low K-values. Figure 11: Convergence of Le// and eqj versus sampling. 32

0.36 - 0.34 I - ~ ~ 0.~32- ~ \~-..01 _SJ~~~~~ ~- ~~~~~~~ ~ - ~.02 0.30 ---.03 - *.04 0.2.8.-.a-.05 0.28 0.26-.,,,,,,,,, 0 200 400 600 800 NSTOP a. The convergence of L.~f on n depends on Is, but is satisfied in all cases considered after 500 terms have been added. 0.350 0.325 0.250- 0, —.-e —-.01 0.275 -.04 0.250 0.225' 0.200. 0 200 400 600 800 1000 1200 MSTOP b.The convergence of LI/ on m is also satisfied after 500 terms. Figure 12: Convergence of Le// on n and m. 33

0.40 0.36 OPTMUM SAMPLING REGICN 0.32 0.28 0.24 NSTOP/1.5 a NSTOP/4a 0.20 0 25 50 75 100 125 150 NX Figure 13: The optimum sampling range is seen to correspond directly with the flat convergence region for the LeF/ computation. 34

I. INTRODUCTION This is the second of two papers concerned with the study of shielding effects in microstrip discontinuities. The companion paper develops a new theoretical method for the full-wave analysis of shielded microstrip discontinuities. The effects of shielding are important in two situations. The first is when the frequency approaches or is above the cutoff frequency for higher order mode propagation. The second occurs when the metal enclosure (Figure 1) is physically close to the circuitry (proximity effects). These effects have not been adequately studied in the past, and are not accounted for in the discontinuity models in most available CAD packages. In addition to improved theoretical methods, there is a great need for experimental data. Published experimental data on microstrip discontinuites is very limited, especially for high microwave (above X-band) and millimeter-wave frequencies. Such measurements are not trivial, but are essential for verification of the theoretical method. This need motivated the experimental study discussed here. This paper (Part II) uses the previously described method (Part I) to study the effects of the shielding cavity on the behavior of one- and two-port discontinuities including open-ends, series gaps and parallel coupled line filters. In addition, comparisons are made to available data from other theoretical solutions including other full-wave analyses and commercially available CAD packages. 1

II. EXPERIMENTAL TECHNIQUES Measured data on microstrip discontinuities is very limited, particularly at higher frequencies (above 10GHz). This is due to the difficulties involved with performing accurate microstrip measurements. In order to measure a microstrip circuit, it is generally mounted in a test fixture with either coax-to-microstrip or waveguide-to-microstrip transitions. The main difficulties associated with such measurements are the separation of test fixture parasitics from measurements, called de-embedding and the non-repeatability of microstrip connections. This section explains the experimental techniques used for this study, and addresses the connection repeatability issues that pertain to the measurements. A. De-embedding Approach The measurement approach of this study employs Automatic Network Analyzer (ANA) techniques in conjunction with the thru-short-delay (TSD) method for deembedding the effects of the test fixture from the measurements. The test fixture that was used is shown in Figure 2. The fixture employs a pair of 7mm "Eisenhart" coax-to-microstrip transitions [1]. The shielding is provided by placing U-shaped covers on top of the microstrip carriers. This forms a cavity similar to Figure 1. The instrumentation used for the measurements was an HP8510 ANA. The test fixture invariably introduces unwanted parasitics and a reference plane shift to the measurements. These effects must be accurately accounted for and removed from the measurements, or incorporated into the ANA system error model. Conventional ANA calibration, which uses a short circuit, an open circuit, and a matched load is not easily performed in microstrip since these calibration standards 2

are much more difficult to realize in microstrip. The process for removing test fixture effects is called de-embedding and consists of two steps: 1) fixture characterization, and 2) the extraction of fixture parasitics. Through de-embedding, the effective calibration reference planes are moved from the coaxial or waveguide ANA test ports to microstrip test ports within the fixture. A comparison of various de-embedding techniques [2] lead to the choice of the TSD technique for the experimental study. This approach was selected over the alternatives considered because the standards used for fixture characterization are the easiest to realize in microstrip, and because the connections to these standards can be made in the same way as the connections made to discontinuity test circuits. In the TSD technique, two-port measurements made on a thru (zero length delay) line, a "short" circuit, and a delay line provide enough information to characterize the fixture. Since the original paper [3], it has been pointed out that the "short" implied in TSD, need not be perfect. In fact, any highly reflecting standard may be used in its place [4,5]. The only requirement is that the same reflection coefficient r, must be presented to both microstrip test ports. This measurement approach provides for the measurement of the effective dielectric constant, the reflection coefficient of open-end discontinuities, and the two-port scattering parameters of series gaps and coupled line filters. In the present implementation of TSD de-embedding, an open-ended microstrip line is used in place of the short as the reflection standard. Measurements of microstrip effective dielectric constant eff, and the reflection coefficient of the open-end rP are obtained as byproducts of the fixture characterization procedure. Once the fixture is char3

acterized, the de-embedded S-parameter measurements of two-port discontinuities are obtained by extracting the fixture parasitics mathematically. B. Connection Repeatability Issues One drawback to the TSD technique is that good microstrip connection repeatability is important for accuracy. Microstrip connections are much harder to make, and less repeatable than connections in coax and waveguide. This is a key limiting factor to the accuracy of microstrip measurements at higher frequencies. To address this issue, a microstrip connection repeatability study was carried out [6]. The results of this study were used to decide on the best connection approach to use and to estimate the associated measurement uncertainties. There are three basic connection alternatives for TSD characterization of a coaxial fixture. Each of these must rely on at least one of the following assumptions: 1. repeatability of connections made from the coax-to-microstrip transition to the microstrip line 2. repeatability of microstrip-to-microstrip interconnects 3. uniformity of electrical characteristics between different transitions (launcherto-launcher uniformity). The results of the repeatability study favor a connection approach relying on repeatable coax/microstrip connections, and this was the approach adopted for the present work. As part of this work, a method was developed to approximate the uncertainties in de-embedded results arising from connection repeatability errors [2]. The anal4

ysis consists of perturbing the S-parameters of the TSD standards and the D.U.T. with a set of experimentally derived error vectors that are representative of the variations of each S-parameter (S1S, S12 etc. ) measurement with repeated connections. Software was written to allow processing the perturbed S-parameter data in the same way as the measurement data is processed during the TSD de-embedding procedure discussed above. This perturbation analysis, shows approximately how connection errors -which are inevitable- propagate through the TSD mathematics and limit the precision of the final results. III. NUMERICAL AND EXPERIMENTAL RESULTS In this section, the numerical and experimental results of the present research are presented for the network parameters of shielded microstrip discontinuities. Included here are results for the effective dielectric constant, open-end and series gap discontinuities, and coupled line filters. Where possible, comparisons are made to results generated from the commercially available CAD packages Super Compact and Touchstone1. The CAD models used in these packages are based on a combination of different theoretical techniques, most often embodied in simplified closed form solutions, curve fit expressions, or look-up tables. These models do not adequately account for the effects of the shielding box (Figure 1). Further, in simulating a circuit containing many discontinuities, the analysis of these packages assume that the discontinuities are independent of one another and the matrix representations for each discontinuity are simply cascaded together mathematically. 1 Super Compact and Touchstone are microwave CAD software pack - ages available from Compact Software and EESOF respectively. 5

In contrast, the full-wave solution presented in Part I accurately treats the entire geometry of the shielded microstrip circuit as a boundary value problem. The interactions between the discontinuity structure, adjacent microstrip conductors, and the shielding cavity are automatically included in the analysis. Because of this, the method is expected to provide better accuracy than CAD model predictions. A. Cutoff Frequency and Higher Order Modes One case where shielding effects are noticeable is when the frequency approaches the cutoff frequency f, for the first higher order shielded microstrip mode. The nature of higher order modes in shielded microstrip is quite different from that in open microstrip. In open microstrip, higher order modes occur in the form of surface waves and radiation modes. The first surface wave mode has a cutoff frequency of zero. In shielded microstrip, the higher order modes take the form of waveguide modes [7]. As a consequence, below the waveguide cutoff frequency, only the dominant microstrip mode can exist. For the present work, the fc for the shielded microstrip geometry of Figure 1, is approximated by considering the dielectric-loaded waveguide formed by removing the strip conductors and the walls at x = 0, and a. The cutoff frequencies so derived have been found to give a good prediction of where higher order effects are first observed in the computed current distributions. As an example, Figure 3 shows the current distribution on an open-ended line operating below the cutoff frequency. For the indicated geometry, f, is about 17.9 GHz. As the frequency is raised above the cutoff frequency, the current becomes more and more distorted as shown in Figure 4. The distortion is due to the interactions between the dominant 6

mode and the first higher order waveguide-like mode inside the cavity. B. Effective Dielectric Constant Figure 5 shows eqff for a 25 mil thick alumina substrate where the cross sectional shielding dimensions, b and c, are ten times the substrate thickness (h). The numerical results are compared to measurements, and to CAD package predictions. Note that Super Compact allows only the cover height to be varied while the calculation provided by Touchstone neglects shielding effects. For the shielding geometry used here, it is seen that the difference between the numerical and CAD package results are within experimental error. However, interestingly enough, better agreement between the CAD results and the numerical results is observed at higher frequencies. This may be due to the fact that the side walls, which are not included in the CAD package analysis are electrically closer to the strip at low frequencies. The measured data is obtained as a byproduct of the TSD fixture characterization procedure as discussed above. The data shown represents the average of ten separate procedures conducted over a period of time with four different sets TSD standards. The error bars shown in Figure 5 represent the standard deviation (~s) of the different measurements. This data is shown here in lieu of the result from a single measurement, since it gives a more representative view of the involved measurement uncertainty. In this case the error bars shown represent the combined effect of connection errors, variations in e, and other factors. The major error source in this case is believed to be the variations in e, which can be significant 7

Table 1: CAVITY NOTATION USED TO DENOTE DIFFERENT GEOMETRY AND SUBSTRATE PARAMETERS CAVITY e, W ( h (in) b ( in) c (in) fc (GHz) CA 9.7.025.025.250.250 21.8 CC 9.7.025.025.100.100 37.5 CF 9.7.025.025.075.075 41.7 QCB 3.82.0157.010.122.080 45.8 QCE 3.82.0157.010.100.100 73.0 QCG 3.82.0157.010.050.05 102.5 for alumina substrates [8]2. To see how Ceff varies with shielding, consider the plot of Figure 6. This plot compares numerical and Super Compact results for three different shielding geometries. The notation used to describe different shielding and substrate geometries is explained in Table 1. In all cases, as the shielding is brought closer to the microstrip a reduction in ~eff is predicted. The case for cavity CA is the same as that of Figure 5. For the other two cases, where the shielding is closer to the microstrip, the Super Compact shows a smaller effect than the present integral equation method predicts. The effect of shielding on eeff for a quartz substrate is displayed in Figure 7. In this case the Super Compact analysis is seen to give good results for both of the two larger shielding geometries. However, the numerical results again show a larger reduction in e1ff as the size of the shielding is decreased further. The reduction of the effective dielectric constant, relative to Super Compact, 2This error reflects the uncertainty of not mknowing the exact value of c, to use in the theoretical simulations. 8

can be explained-as follows. For a larger shielding geometry, the field distribution on the microstrip more closely resembles the open microstrip case, with most of the electric field concentrated in the substrate. In this case, most of the electric field lines originate on the microstrip conductors and terminate on the ground plane below. As the cavity size is reduced, the ground planes of the top anestd side-walls are brought closer to the microstrip lines. The electric field distribution is now less concentrated in the substrate, as more field lines can terminate on the top and side walls. As a result, a proportionally larger percentage of the energy propagating down the line does so in the air region, and the dielectric constant is reduced. C. Open-end Discontinuity As discussed in Part I of the paper, an open-end discontinuity can be represented by an effective length extension Leffj or by a shunt capacitance cp. Both of these three representations will be used in this section. The plot of Figure 8 compares Lcff results to those of Jansen et al. [9] and Itoh [10]. In this case, the dimensions of the shielding cavity are large with respect to the substrate thickness. The results from this research are almost identical to those obtained by Jansen et al. for frequencies above 8 GHz, but show a reduced value for lower frequencies. The case of Figure 8 was chosen to compare the coaxial and gap generator excitation methods used in the method of moments solution Table 2 shows that the results computed for this case by the two methods are equivalent. This equivalence also holds for the two-port scattering parameters for the structures considered herein. Hence, as far as computing network parameters is concerned either 9

Table 2: COMPARISON OF Lef1/h COMPUTATION FOR THE TWO TYPES OF EXCITATION METHODS f (GHz) 4 8 12 14 16 18 20 GAP GENERATOR.298.305.309.321.324.344.353 COAXIAL EXCITATION.299.304.309.322.327.344.352 method gives good results. Since the coaxial method is more realistically based, this conclusion lends validity to the use of the gap generator method. The results shown in Figure 9 illustrate the effect of shielding on the openend discontinuity. The normalized open-end capacitance cop is plotted for three different cavity sizes. The results show that reducing the cavity size raises fc (as expected), and it lowers the value of co. For comparison, data obtained from Super Compact and Touchstone and measurements (see Section 4.3) are included. The errors bars on the measurements represent the estimated standard deviation (~s) of the connection errors associated with this measurement3. Similar shielding effects are observed for an open-end on a quartz substrate as shown in Figure 10. In this case it is seen that the Super Compact result gives a good value for low frequencies, and where the frequency is well below the cutoff frequency for a given shielding size. These results show that shielding effects due to wall proximity are less important than shielding effects due to the onset of higher order modes. 3The other error sources indicated for the effective dielectric constant measurement are not considered to be as significant for this measurement. 10

D. Series Gap Discontinuities Numerical and experimental results have been obtained for series gap discontinuities of three different gap widths [2]. Results for one of these gaps are presented here. Numerical results for the magnitude of S21 of a series gaps with a 15mil gap width are shown plotted in Figure 11. For comparison, results obtained using Super Compact, and Touchstone are also shown plotted along with measured data. The numerical results are seen to be in very good agreement with the measurements. The test substrate and shielding dimensions used for the measurements are those for cavity CA (Table 1). The error bars associated with the connection errors, are on the order of ~.5dB and are too small to show on the plots. Results for the angle of S21 and S11 for the 15 mil series gap are shown in Figures 12 and 13. The error bars in these charts represent the estimated standard deviation from the perturbation analysis4. Although the measurements tend to favor the numerical results, the phase differences are not too significant since it is suspected that the measurement may be in error by more than that attributed to connection errors alone. The phase of the S-parameters for the other two gaps behave in a similar way as that for the 15 mil gap and have been omitted from this treatment. These results are seen to further verify that the theory developed in Part I. For the large shielding dimensions used for the measurements (b, c ~ h) the CAD models are also seen to give reasonable predictions. The behavior of series gaps 4The analysis was caied out at 100GH, and it is amumed that the connection error ar approximately the same at the other measurement frequencies. 11

for different shielding dimensions was not studied, instead emphasis was placed on obtaining results for coupled line filters since their behavior is more complicated and therefore more interesting. E. Four Resonator Coupled Line Filter The last results to be presented are for the four resonator coupled line filter of Figure 14. For brevity, only the amplitude and phase of S21 will be discussed. Numerical and measured results of this research are compared along with CAD model predictions in Figure 15. The CAD package analysis for coupled line filters is performed by cascading two different types of discontinuity elements together: coupled microstrip lines, and open-end discontinuities. Neither of the packages studied here account for shielding in the open-end discontinuity model, however, Super Compact does include the effect of the cover height in the model for coupled lines. The numerical results shown in Figure 15 demonstrate excellent agreement with measurements up to the cutoff frequency. The cutoff frequency f, for the shielding geometry of the filter is approximately 13.9GHz. Above this frequency, the measurements are distorted due to waveguide moding within the test fixture. The results of Figure 15 show that even for large shielding dimensions the discrepancies are apparent in the CAD model predictions, whereas the numerical results follow the measurements closely, both in amplitude and phase. As can be seen from the amplitude response (Figure 15a), the CAD models give a good prediction in the pass band, but fail to predict the filter response in the rejection band. This is also seen from the phase response (Figure 15b), where the CAD 12

models display a large error compared to measurements, between about 6 and 8.5GHz, while the numerical results track the measured amplitude and phase very well. Below about 5.5GHz, the measured phase is seen to be different from the predictions of both the CAD models and the numerical results. This is most likely due to a phase error in the measurements. In the TSD technique, the delay line for the measurements should ideally be A at the measurement frequency 5. When the electrical length becomes either too short or too close to a multiple of Af phase ambiguities can result. A good rule of thumb is for the delay line to be between A and 3-. At 5.5GHz the delay line used for the measurements is slightly less than AA; hence, this is most likely the source of the phase error in the measurements below this frequency. We will now examine what happens as the top cover is brought closer to the circuitry. Figure 16a shows Super Compact predictions for the four resonator filter with two different cover heights. These predictions indicate that lowering the cover height should significantly narrow the pass band, and reduce the amplitude in the rejection band. A significantly different prediction is observed in the numerical results for this case presented in Figure 16b. A narrowing of the pass band response is also observed in the numerical predictions, but not by nearly as much as in the Super Compact predicts. More importantly, the amplitude in the rejection band is seen to increase instead of decrease. SMultiple lines are needed for broadband measuremts. 13

To prove that-the numerical prediction is indeed the correct one, an additional measurement was made of the filter for the low cover height case. As can be seen from Figure 16b the agreement between measured data and the numerical predictions from this research is excellent. IV. SUMMARY AND CONCLUSIONS In this paper theoretical and experimental results were presented for the network parameters of one- and two-port discontinuities. For the measurements, the TSD de-embedding approach was used. Connection repeatability errors were considered in detail and a perturbation analysis was developed to approximate their effect on the precision of the final de-embedded results. The effects of shielding on microstrip behavior was studied. It has been demonstrated that the computed current distribution becomes distorted above the cutoff frequency fcfor the first higher order shielded microstrip mode. On the other hand, as long as the cavity size is such that the frequency is below fc, the current is uniform and undistorted regardless of how thick the substrate is. Only one of the CAD packages studied takes shielding into account for the effective dielectric constant (eiff) calculation, and then only cover effects are considered. A comparison of the CAD package predictions with the numerical results of this research for eff showed that good agreement is obtained when the shielding dimensions are large with respect to the substrate thickness, while for small shielding dimensions, the difference between the different results becomes significant. For the open-end discontinuity, good agreement with other full-wave solutions and with measurements has been demonstrated. A comparison of open-end capac14

itance for different cavity sizes showed that, as the cutoff frequency is approached, the capacitance increases in each case. Choosing a small cavity with a high cut-off frequency extends the region where the capacitance is relatively constant. Good agreement between numerical and measured results was also demonstrated for series gap discontinuities and a four resonator coupled line filter. For the filter, reducing the cover height was seen to narrow the pass band response and raise the amplitude of the filter's rejection band response. The numerical results of this research give an excellent prediction of this effect, whereas discrepancies are apparent in the CAD model predictions. ACKNOWLEDGEMENTS The authors thank Mr. Ed Watkins, Mr. Jim Schellenberg, and Mr. Marcel Tutt for their contributions to this work. This work was primarily sponsored by The National Science Foundation (Contract. No. ECS -8602530). Partial sponsorship was also provided by the Army Research Office (Contract No. DAAL0387-K-0088) and the Microwave Products Division of Hughes Aircraft Co. References [1] R.L. Eisenhart,"A Better Microstrip Connector", 1978 IEEE MTT-S Digest, pp. 318-320. [2] L.P. Dunleavy, "Discontinuity Characterization in Shielded Microstrip: A Theoretical and Experimental Study", Ph.D. dissertation, University of Michi15

gan, April 1988. [3] N. R. Franzen and R. A. Speciale, "A New Procedure for System Calibration and Error Removal in Automated S-Parameter Measurements", 5th European Microwave Conference,1975, pp. 69-73. [4] B. Bianco et. al. "Launcher and Microstrip Characterization" IEEE Trans. on Instrum. and Meas. Vol. IM 25, NO. 4, Dec. 1976, pp. 320-323. [5] G. Engen and C. Hoer, "Thru-Reflect-Line: An improved Technique for Calibrating the Six-Port Automatic Network Analyzer". IEEE Trans. Microwave Theory Tech. vol. MTT-27, No.. 12, Dec. 1979, pp 987-993. [6] L. Dunleavy and P. Katehi, "Repeatability Issues for De-embedding Microstrip Discontinuity S-parameter Measurements By the TSD Technique" Automatic RF Techniques Group (ARFTG) Conf. Dig. June 1986. [7] E. Yamashita and K. Atsuki, "Analysis of Microstrip-Like Transmission Lines by Nonuniform Discretization of Integral Equations," IEEE Trans. Microwave Theory Tech., vol. MTT-24, pp. 195-200. [8] J. Snook, "Substrates for Hybrid Microelectronic Applications", Microwave System News and Comm. Tech., February 1988, pp26-31. [9] R.H. Jansen, and N.H.L. Koster, "Accurate Results on the End Effect of Single and Coupled Lines for Use in Microwave Circuit Design " A.E.U. vol. 34, pp 453-459, 1980. 16

[10] T. Itoh,"Analysis of Microstrip Resonators", IEEE Trans. Microwave Theory Tech., vol MTT-22, pp 946-951, 1974. 17

microstrip shielding cavity (or housing) coaxial I coaxial input > / 1 output Z=C Figure, 1: Buic geometry for the shielded mic trip caviy z~....'':'.::.'...S.'./^. z=hfB / Ix / x=a dielectric substrate Figure 1: Basic geometry for the shielded microstrip cavity problem. 18

Figure 2: 7mm coaxial/microstrip test fixture (partially disassembled) used for measurements. 19

0.874E- AA 0.523E-03 o. I 73E-03 - -.178E-03 -0.528E-03 -0.879E-05 3 I 0.000 1.275 2.550 3.825 5.100 6.375 X(WAVELENGTHS) Figure 3: Below the cutoff frequency fc, the microstrip current on an open-ended line forms a uniform standing wave pattern (f = 16GHz,e, = 9.7, W/h = 1.57, h =.025", b = c =.275"). 20

0.49 E-03 0.300E-03 0. 1 03E-03 -0.937E-04 -0.290CE-03 -0.487E-03 0.000 t.750 3.500 5.250 7.000 8.750 X(WAVELENGTHS) Figure 4: As the frequency is increased above f,, the current becomes more and more distorted due to higher order modes in the current distribution (f = 22GHz,c, =..9.7.W/.h-=.5.t.. =.025",b = c =.275"). 21

7.6 7.4 7.2: 7.0, SUPER COMPACT C< 6.8 TOUCHSTONE B — THS RESEARCH 6.6 g4; 6.4T J- t MEASUR6BEENT 6.2 6.0 0 2 4 6 8 10 12 14 16 18 20 22 FREQUENCY (GHZ) Figure 5: The numerical results for ec.t show good agreement with both measurements and CAD package results for an alumina substrate with large shielding dimensions (e, = 9.7, h =.025", b = c =.25"). 22

8.5 8.0 7.5 -y A — ~. CA: SUPER COMPACT,A J^-A C-C- CC: SUPER COMPACT 7.;' --- CF: SUPER COMPACT;co 7.0, - f>.* CA:THIS RESEARCH *~z v|CC: THIS RESEARCH 6.5 A A ~ CF: THIS RESEARCH 6.0 A 5.5. -,, - I, I 0 8 16 24 32 40 48 56 64 FREQUENCY (GHZ) Figure 6: The effects of shielding on el/l are apparent as the size of the shielding cavity is reduced (see Table 5.1 for geometry.) 3.3 3.2 LA 31.' —-- QCB: SUPER COMPACT * n*-Qi QCE SUPER COMPACT aj 3.0 * QCG: SUPER COMPACT.^ *A * OC8: THIS RESEARCH." ~* E * QCTHSRESEARCH 2.-~~~9"]~ yff^ ArA Q CG:THS RESEARCH A 28. 7 & 2.7..,.,.,., 0 20 40 60 80 100 120 FREQUENCY(GHZ) Figure 7: Shielding effects are also significant for the quartz substrate shown here (see Table 5.1 for geometry). 23

0.45 0~40- a 17.9GHZ 0.40 0.35 0.30 _ ""^yi ll ^l ^ -U-~- JANSEN & KOSTER _ 0.30 l- * THIS RESEARCH ~ -IOH 0.25 0.20 0.15.. 0 4 8 12 16 20 24 28 FREQUENCY (GHZ) Figure 8: The numerical results of this research are also seen to compare well with other full-wave analyses (e, = 9.6, W/h = 1.57, b =.305", c =.2", h =.025"). 24

3.5 3.0 _ * f',-20.8GHZ E 2.5 | -- SUPER COMPACT o~~ i / _^ -ri 8 - @- - r TOUCHSTCNE *S~~~2~~~. go-YA~ ff0iff11~ ~- CA. THIS RESEARCH _ / - CC: THIS RESEARCH I-' -aal^^^^ r^ — * CE:THI RESEARCH Ot 1.5 l | i CA: MEASUREMENT f,-41.7GHZ 1.0 f, -37.5GHZ 0.5 0 10 20 30 40 50 60 70 FREQUENCY (GHZ) Figure 9: A comparison of the normalized open-end capacitance for three different cavity sizes shows that shielding effects are dominated by the onset of higher order modes rather than by proximity effects (see Table 5.1 for cavity geometries). 25

0.8 f- 73.0GHZ c 73HZ f 102.SGHZ 0.7 - f~ 45.8GH2 0.6 — U-"='- SUPER COMPACT r - *CE THS 0| i C: THIS RESEARCH FREQUFie 10: No d on-ed ce fr te d t c y ss fr a qZ alsoFigure 10: Nomalizeng dependend capacitance for three di5.1 for cavity geometries for a quartz substrate also sho26s a strong dependence on f (see Table 5.1 for cavity geometries). 26

0.0 -7.5.1 5.0 -,,saCw9' -i-.0 TOUCHSTONE.22.5 _%'22'5,^~^ ^ ^ ~^a- SUPER COMPACT u -30.0 - ^ * THIS RESEARCH 0 -30.0 N i MEASUREMENT -37.5 -45.0 - |. I * I. I 0 2 4 6 8 10 12 14 16 18 20 22 FREQUENCY (GZ) Figure 11: The numerical results show good agreement with measurements for the magnitude of S21 for a 15mil series gap. 27

95 85 ~' cotIT-~ TOUCHSTONE 75 T I --- SUPER COMPACT UWs=~~~~~~~~~ ^^ T — -- TIS RESEARCH v 65 4I MEASUREMENT 65 55 i,,, 0 2 4 6 8 10 12 14 16 18 20 22 FREQUENCY (GHZ) Figure 12: Angle of S21 for series gap. 28

. -lo =I 3a -.10*'-* 8 i- TOUcHSTONE W = SUPER COMPACT - s i M —RESEMCH -320. ~. MEASUREMENT -30- i............. 0 2 4 6 8 10 12 14 16 18 20 22 FREQUENCY (GHZ) Figure 13: Angle of S11 for series gap. 29

a, I I I 1 I I i I W = 11.9 1 1. I I Figure 14: Sketch of four resonator coupled line filter studied here ( er = 9.7, h-=;-.025", b-=A", c =.25"). I I30 I I I I I $ I * I I 2 I I I I I I I I I I I a''L! I Figure 14: Sketch of four resonator coupled line filter studied here ( er = 9.7, h —';.025,; hb-.4", 30

-20, ^40. D -, --- TOUCHSTONE J r 4 —ll SUPER COMPACT _~^ ~. -*~ _~ / ^~-m -~- * THS RESEARCH 43'60' JT / | MEASUREMNIB -80 IY fc -13.9GHZ -100 2 4 6 8 10 12 14 16 18 20 FREQUENCY(GHZ) a. Amplitude of S21 200 100 all Q ~~~~~~-a- TOUCHSTCNE FREQ Y (MEASUHRBANT -200 2 4 6 8 10 12 14 16 18 20 FREQUENCY (GHZ) b. Phase of S2 Figure 15: Results for transmission coefficient S21 of four resonator filter (e, = 9.7,W =.012", h =.025"; =.4, c =.25"). 31

d \n |~ ~ SUPER COMPACT -20 FESULTS 1, i, l -- c.25' -40 - -80". t' ~ \..... -.. 7." <o-60- /i c07 -100 2 4 6 8 10 12 14 16 18 20 FREQUENCY(GHZ) a. Super Compact predictions NUMERICAL DATA -20 1 0*u~ *|'. 2 ~~ * "..- c.075' ~-40 0 -" c =.25. -100 -I 2 4 6 8 10 12 14 16 18 20 FMEAREE4YD DA(TA b. Numeric lts of th arch compared to measurements13 Figure 16: Results for lowering the shielding cover on the amplitude response of four resonator filter (e, = 9.7, W =.012", h =.025"; b =.4"). 32 32

Appendix D "High Frequency Conductor Losses in Shielded Microstrip" T.E. van Deventer and P.B. Katehi 14

ARO - 024562-2-T HIGH FREQUENCY CONDUCTOR LOSSES IN SHIELDED MICROSTRIP T.E. van Deventer P.B. Katehi Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109 June, 1988

1 Introduction The effect of conductor losses in microstrip circuits and especially MMIC's is important to the circuit designer who has to account for dissipation and power loss. Several studies have been performed to calculate the dispersion in microstrip structures at microwave frequencies accounting for dielectric and conductor losses. Conductor losses have first been analyzed by Pucel [1] and Wheeler [2] where a technique based on the incremental inductance rule was used. Other approaches to the problem include quasi-TEM models [3], [4] and the conventional perturbation technique as in [5] using a spectral-domain approach. In the present study, a shielded planar microstrip transmission line is considered where both dielectric losses in the substrate and conductor losses in the strip are accounted for. A spectral approach is adopted to solve an equivalent problem where the dielectric layer (or layers) is replaced by an impedance boundary condition. Using Fourier transformation, the Green's function of the problem is derived. Assuming a thin-strip approximation and a Maxwellian distribution for the current, a method is applied that represents conductor losses in microstrip lines in terms of a frequency-dependent impedance. The propagation constant of the lossy line is then computed for different frequencies and loss tangents. 1

2 Derivation of the Green's function 2.1 Geometry Consider an infinitely long inhomogeneously-filled waveguide, with a microstrip centered on the substrate as shown in Figure 1. The dielectric substrate is considered lossy with relative permittivity r, and permeability i,. Conductor losses are accounted for in the microstrip whereas the walls are assumed perfect conductors. Striplines which are within a shielding structure that is completely filled with dielectric material can propagate TEM waves. Shielded microstrip lines, however, cannot support these modes because the boundary conditions at the interface between air and dielectric cannot be rigourously fulfilled. These lines propagate hybrid modes which have non-zero cut-off frequencies with the exception of the dominant mode. Each microstrip mode propagates rectilinearly along the z-direction. Therefore the z-dependence in an infinite line will be of the form e-'"z where 7% is the complex propagation constant for the given mode. The solution for the microstrip modes can be obtained through an Az, Fz or A, F formulation which has to be applied to each of n dielectric regions separately [6]. This approach leads to a linear set of 4(n+1) equations which has to be solved analytically in order to find the unknown potentials. In the present formulation we eliminated this limitation by considering an equivalent boundary condition on the air-dielectric interface. In this manner, we are able to decrease the complexity of the solution and solve for the electromagnetic fields in the air region only. The impedance boundary condition is applied separately for LSE and LSM modes. The current source is raised above the substrate interface to study the slightly more 2

x a 2 2W ~I L o h~. ~.... ~:~!-.. ***-,..*A.......... Yo b Figure 1: Cross-section of the shielded microstrip line x X (I) a point source 0 (II) impedance layer Yo Y b Figure 2: Cross-section of the equivalent boundary problem 3

general type of configuration as shown in Figure 2. Note that the height of the waveguide is now a', where a' = a - h. 2.2 Derivation The Maxwell's equations are applied as V x = -jwHf (1) V x H J + jwE (2) V. (E) =p (3) V. (D) = 0. (4) Defining the electric and magnetic potentials in the usual way using the Lorentz condition, we get for LSE modes F = F x (5) E= V x F (6) H = jwF- - VV F, (7) and for LSM modes, A = Ax (8) - 1- - H -V x A (9) E = -jwA+ VV * A. (10) The problem is Fourier transformed using a spectral domain approach to reduce the problem to a two-dimensional one by eliminating the z-dependence. 4

The transformations used are given by A, = 2 Ax, ekz dkz (11) A, = A e-jkz dz. (12) J-00 This gives for LSE modes E = 0 (13) Ey = j F (14) E Ez = -- -F (15) 1 2 Hx = ---— (k2-k)F (16) Hy = j1 02a (17) Hz = -Fx (18) WEm OXa and for LSM modes, H, = 0 (19) Hy = jA, (20) Hz = —AX (21) Ex= (k- k) Ax (22) Ey 1 02( Ey jwep yA (23) -E, - - A (24) WE/I Ox The general solution to the wave equation in each region can be obtained by the 5

method of separation of variables as 00 A') amn cos(k=2(x - a)) sin(b (25) (I=) E [bmncos(kx2X)+ Cmnsin(kz2)] sin( b ) (26) F( = E dn sin(k2(x- a)) cos(mb ) (27) m 00mry () = Z [fmn cos(k2xr) + gmn sin(kax)] cos(mb) (28) where kxi and kC2 represent the wave numbers in the dielectric layer and in the air region respectively and (I), (II) correspond to the air region above and below the point source respectively. The boundary conditions on the perfectly conducting walls have been used to derive these expressions. The problem then consists of solving for the six unknown coefficients amn, bnn, Cmn, dmn, fmn and gmn using the following boundary conditions on the x = x0 interface in the Fourier domain. H1I) - H) =5(y - y')e-jkZ (29) E() = E(I (30) E( = Ezi (31) Hz) = H(I). (32) Also impedance boundary conditions are used on the interface as (I LSE ffL\ l) (33) Y-(II)) - 7m (34) These impedances r7e and rim can be found using the sending end impedance formulas for transmission lines. For a single layer these are, for LSE and LSM modes 6

respectively Te -= k tan(kilh) (35) kl1 r7im = -j2 tan(klh) (36) W~d After solving for the six coefficients, one obtains for the electric field due to a dipole above a substrate vll II m 7/ 1 7 Mr I. /mr / Ez = E m sin(k2(Xo - a')) + sin(T Y') sin(am y)e-k m=l J b b m W2 we k.2 sin(kxlh) cos(kT2x) + kl1 cos(kxlh) sin(kx2x) b k2 cos(kx2(xo - a'))(kx2 sin(klh) cos(k,2xo) + kxl cos(cklh) sin(kx2xo) 1 + sin(kc2(Xo - a'))(kx2 sin(ksilh) sin(kT2Xo) - ks1 cos(k.lh) cos(kc2Xo)) klkx2k 2 Erkx2 sin(kTz2x) cos(kxlh) + kIl cos(kT2x) sin(kclih) WL - cos(kca2(XO - al))(Erkx2 cos(klih) sin(kx2Xo) + k1l sin(kclh) cos(kT2xo _______________________________). (3 + sin(kc2(Xo - a'))(ka2Er cos(kclh) cos(ka2Xo) - kc1l sin(k)ih) sin(kz2xo)) When restoring the point source on the interface x = Xo, the electric field in (37) therefore becomes piJZ * 2.m0r, M. kmr z = -jwlL- 1 sin(k2a') sin(iy) sin( b y)e-jkz' m-1 m(T)2 k 1 ( (.L)2 sin(kxlh) kz + (m )2 kx2 sin(kxlh) cos(k2ka') + kx1 cos(cklh) sin(k2a') k 2k,2kz,l sin(klh) (38) ko2 k1 sin(kxlh) cos(kx2a') + k26re cos(klh) sin(kx2a') The denominator of each of the two terms inside the brackets is seen to correspond to the transcendental equations for LSE and LSM modes. Therefore we can write the E fields as Ez)lIx=xo = EILSE + EzILSM (39) 7

The equation (38) corresponds to the Fourier transform of the Green's function of the problem, namely G^. Its inverse Fourier transform as defined in (11) is then given by G = 2- Gzz ejkz dkz. (40) 27r J-0o 2.3 Electric field on the strip Assuming that the strip is thin enough in the y-direction, we can propose a Maxwellian distribution [8] for the current density on the microstrip as J(x', y' z') = 2(x' - xo) ejkisz, (41) lrW 1-2 [ -Yo)].b/1_-[2(y'-,_ Y0) where w is the width of the line and kMS is the unknown microstrip propagation constant. The function f (y) allows for single or multiple strips on the air-dielectric interface. Chebyshev and Legendre polynomials have been used recently for the current distribution [7]. This study will first focus on a single microstrip centered in the y-direction. In that particular case f(y) reduces to unity. The electric field due to this current distribution has a variation in x', y' and z' where the corresponding integrals are: = S(x'-xo)dx' (42) Yfo+f m', a 1M Iy = - sin(y) 1 dy (43) Jyo-j a v 6 b 2 I o/, p" o2 sn —I = ejkMSz' eJklz-z' dz'. (44) J-oo The goal of this study is to find the dispersion characteristics of a centered, narrow microstrip line. To this end, the electric field on the interface x = xo is calculated 8

as follows E,(x= xo) = ( G(x = xo) J,(y', z') dy' dz'. (45) Substituting (38) into (45), we get vl(=21) \ ~n1 I'"+T~ p0 />+0 2 +00 l\ * /m7r m7r E~(x = xo) = - /I2 L 1i -jw sin( ska)s( ( y) Jyo- J-oo J-oo M1 b 1i. ( _ (mb()2 Sin(kxlh) k2 + (mZK)2 k2 sin(kxlh) cos(kx2a') + kxl cos(klh) sin(k2a') k2 kx2kxl sin(kilh) ko kxl sin(kxlh) cos(kx2a') + kx2er cos(klh) sin(kx2a') 2 1 ejkJSz'e-jkz' dy'dz'dk,. (46) 7 -W 1- [u('- yo)]2 or more simply E1 = fo+o >+ +oo m7 — Ez(x = xo) = 2 (y, m, kl, kI2, k) sin( y') 2-r = -o -oo o 2 1 e-j(kz-kMs)X dy' dz' dkz. (47) w 1- [2(y' — o)] Evaluating the integration in y' (see Appendix F, [9]), one gets 1 m +oo 0+oo +i Mr Ez(x x0) = 2 12 q(y, m,k 1,k2, k) sin( - yo) 1r m=1 — 00 00 b mr w (kz Jo( -— ~) ej(kk S)z' dz' dkz (48) 2 ( +y, m,,l,,, k2) kz)sin( mYo) r m=1 00 b Jo(mt)i(k, - kfs) dkz (49) 2 1r y](Y mly m, k~,2, k,) sin( mrO Jo( mrb )|I,=kMs (50) (51) 9

or,(z ~ \~ ^1 -o 2 mr Ez(x=xo) = 1 -J.2sin(k.2a)s( y) 27r - n(k2a)sin( Y) 1 ('m%)2sin(kxlh) kz + (Ti)2 s2 sin(klh) cos(kx2a') + klC cos(kclh) sin(kx2a') kz2k12kl sin(k.lh) + k k 1 sin(k.lh) cos(kx2a') + x2Er cos(kilh) sin(kx2a') mir mir w sin(- Yo) Jo( — )Ik,|=M (52) It can be noted that only the odd modes in m will add up when the strip is centered in the y-direction. One more condition needs to be applied, i.e. the boundary condition on the microstrip. For perfect conductors, the tangential electric field vanishes on the strip. In this study, we assume the strip to have a finite non-zero surface impedance. This impedance varies both with frequency and space in the y-direction. The width is assumed large compared to the thickness of the strip but the latter need not be large compared to the skin depth. A definition was proposed in [10],[11] for the surface impedance as Z(f, y) = (R + jwLi.)w (53) where R and Lin are the per unit length resistance and internal inductance of the microstrip. By definition, the surface impedance is also given by = -Z(f,y) (54) Hy which implies that Hy= Jz (55) 10

since x Hy = J (56) Thus E, + Z(f,y)J, = 0 (57) The integral equation for the electric field in the z-direction is given by 1 ~+oo fo+a - E, = I - i 2 (kI + VV) G Jdy'dz' (58) j*WoE _-00 yo-fL where J is a z-directed current with amplitude given by (41). To minimize the error function, we integrate over the strip after multiplying by a test function. A pulse function has been used in this study. The problem then amounts to solving the following equation for the microstrip propagation constant at the plane z = 0. ]/Y [E + Z(f, y)J]dy = 0 (59) or * 1 1 m7irw 2 nm r sin(k-,a') sin(k.,Ih) sin(' ) sin('yo) 7r- b 2 b,y)=l m w 1 ____ )2_ ~(' b 2)k + (mb )2 ( 2 sin(kljh) cos(kx2a') + k l cos(klh) sin(k,2a') kz2k,2kl 1 Z2 1 ) Ik=kM + ko kx sin(kolh) cos(kx2a') + k,2E cos(kih) sin(kl2a') kIk=S + Z(f) = 0 (60) where Z(f) represents now an average impedance integrated over the width of the strip. This expression can be easily programed in a personal computer to evaluate the propagation constant of various microstrip modes. 11

3 Discussion Based on the theory derived in the previous section, a computer program has been developed to calculate the propagation constant of a lossy microstrip line. The equation (60) is solved for kMS with as many as a hundred modes to insure convergence. The Muller's algorithm is used to calculate the complex roots of this function. Preliminary data were calculated for the case of a lossy substrate with a perfectly conducting microstrip structure. The results have been found to give good agreement with [12], within the readability of the quoted results. Also it is seen that as one decreases the size of the structure while keeping the different ratios the same, the effect of conductor losses on the propagation constant of the quasiTEM microstrip mode becomes significant. This is due in part to the dominant effect of dielectric losses for thicker substrates. The effect of frequency with no dielectric loss is shown in Figs. 3,4,5,6 and also Figure 7. More research needs to be performed in this area, for example on the effect of conductor losses on higher order modes, and the validity of different current distributions on the strip. 12

7.000 5.727 4.455 3.182 -Conductor Losses (real) -0- Conductor Losses Imag).. 19090-i- No Conductor Losses(real) O 1.909 A- No Conductor Losses (Imag): _ —.% 0.636:fr = 8.875 01.636 =.0 /)O S< a = b = 0.635 mm 2.0.636 h = w = 0.0635 mm f=10 GHz. -1.909 -3.182 -4.455.5.727 -7.000 0.000 1.250 2.500 3.750 5.000 6.250 7.500 8.750 10.000 Loss Tangent Figure 3: Propagation constant vs. loss tangent at f = 10 GIlz 13

7.000 5.727 4.455 3.182 Conductor Losses real) -O Conductor Losses (Imag) -O — No Conductor Losses(real) 0 1.909 r No Conductor Losses (Imag) E, = 8.875 0.636 = 1.0 _l) a = b 0.635 mm i -0.636 h = w = 0.0635 mm f = 25 GHz.1.909 -3.182 -4.455 -5.727.7.000',,,,. I, I,,, I,, I I,, I,.. -7.000!,,, 0.000 1.250 2.500 3.750 5.000 6.250 7.500 8.750 10.000 Loss Tangent Figure 4: Propagation constant vs. loss tangent at f = 25 GHz 14

7.000 5.727 - 4.455 3.182 - Conductor Losses real) -0- Conductor Losses (Imag):D 1.909 -0- No Conductor Losses(real) 0 1.909 - 6 No Conductor Losses (imag) _.66 er,= 8.875 0.636r = 1.0 ) a = b = 0.635 mm:. -0.636 0 -6 h = w = 0.0635 mm f = 50 GHz -1.909 -3.182 4.455 -5.727 -7.000,,,,,,,,, I,, I,,,,,,,,, 0.000 1.250 2.500 3.750 5.000 6.250 7.500 8.750 10.000 Loss Tangent Figure 5: Propagation constant vs. loss tangent at f = 50 GHz 15

7.000, g | I 5.727 4.455 3.182 - Conductor Losses (real) -0- Conductor Losses (Imag) -o No Conductor Losses(real) 0 1.909 — z- No Conductor Losses (Imag) 0.636_ ^~~~~, = 8.875 0.636 r 1.0 (f) a = b= 0.635 mm __ -0.636 h = w 0.0635 mm ~E^~~;^f = 75 GHz -1.909 -3.182 -4.455 -5.727 -7.000 0.000 1.250 2.500 3.750 5.000 6.250 7.500 8.750 10.000 Loss Tangent Figure 6: Propagation constant vs. loss tangent at f = 75 GHz 16

3.000 2.682 - 2.364 2.045 Conductor Losses (real) - Conductor Losses (Imag) -: No Conductor Losses(real) 0 1.727 -A No Conductor Losses (Imag) 1.409_^~~~~~ ~, = 8.875 1.409. r = 1.0 ): a b= 0.635 mm 1.091 h =w = 0.0635 mm'_^~~~ *: tan8 =0 0.773 0.455 0.136 ~ -0.182 -0.500 I I I a 2, I 2. I!., I,. I t. I... 10.000 18.125 26.250 34.375 42.500 50.625 58.750 66.875 75.000 Frequency Figure 7: Propagation constant vs. frequency at tan6 = 0 17

References [1] R. A. Pucel, D. J. Masse, C. P. Hartwig, "Losses in Microstrip," IEEE Trans. Microwave Theory Tech., vol. MTT-16, p 342, June 1968. [2] H. A. Wheeler, "Transmission-line properties of a strip on a dielectric sheet on a plane," IEEE Trans. Microwave Theory Tech., vol. MTT-25, p 631, August 1977. [3] M.V. Schneider, " Microstrip Lines for Microwave Integrated Circuits," The Bell System Technical Journal, pp 1421-1444, May-June 1969. [4] B.E. Spielman, "Dissipation Loss Effects in Isolated and Coupled Transmission Lines," IEEE Trans. Microwave Theory Tech., vol. MTT-25, pp 648-655, August 1977. [5] D. Mirshekar-Syahkal, J. B. Davies, "Accurate Solution of Microstrip and Coplanar Structures for Dispersion and for Dielectric and Conductor Losses", IEEE Trans. Microwave Theory Tech., vol. MTT-27, p 694, July 1979. [6] R.F. Harrington, "Time-Harmonic Electromagnetic Fields," 3rd. ed., McGraw-Hill Book Co., 1961. [7] C.J. Railton, T.Rozzi, "Complex Modes in Boxed Microstrip," IEEE Trans. Microwave Theory Tech., vol. MTT-36, p 865, May 1988. [8] J. C. Maxwell, " A Treatise on Electricity and Magnetism," 3rd. ed., vol.l, New York: Dover 1954, pp 296-297. 18

[9] L. P. Dunleavy, "Discontinuity Characterization in Shielded Microstrip: a Theoretical and Experimental Study", Ph.D. Dissertation, the University of Michigan, April 1988. [10] A.C. Cangellaris, University of Arizona, Personal Communication, April 1988. [11] A.C. Cangellaris, "The importance of skin-effect in microstrip lines at high frequencies", IEEE MTT-International Symposium Digest, p 197, May 1988. [12] D. Mirshekar-Syahkal, "An Accurate Determination of Dielectric Loss Effect in Monolithic Microwave Integrated Circuits Including Microstrip and Coupled Microstrip Lines," IEEE Trans. Microwave Theory Tech., vol. MTT-31, pp 950-954, November 1983. 19

"Theoretical Characterization of an Air-Bridge" M.E. Coluzzi and P.B. Katehi 15

ARO 024562-1-T THEORETICAL CHARACTERIZATION OF AIR-BRIDGES Michael E. Coluzzi Pisti B. Katehi Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109 June 1988

Abstract-The theoretical analysis of an enclosed air-bridge is presented. This includes a derivation of the Green's function in the x and z directions which is used to find the current distribution on the conducting strips of the air-bridge. Boundary conditions at the interfaces are applied and the numerical technique Method of Moments is used to solve the integral equation for the unknown current. Upon derivation of the current distribution on the conductors, an ideal transmission line model is applied to obtain the scattering parameters of the structure. 1

Contents 1 Introduction 2 Evaluation of the unknown current 2 2.1 Formulation of the Integral Equation................. 3 2.2 Derivation of Green's function..................... 5 2.2.1 Derivation of the Green's function for a x-directed current.6 2.2.2 Derivation of the Green's function for a z-directed current. 12 2.2.3 Summary of Green's function determination........ 20 2.3 Application of Method of Moments................. 20 2.4 Matrix Equation........................... 34 3 Scattering Parameters 34 4 Summary 34 2

1 Introduction Millimeter wave technology concerns itself with that portion of the electromagnetic spectrum between 0.3 Ghz and 300 Ghz, corresponding to wavelengths of 1000 mm to 1mm. Effective quasi-static techniques have been developed for the lower frequencies (0.3 Ghz to 3 Ghz) but for the higher frequency part of the spectrum, a full-wave analysis must be employed. Millimeter and microwave systems may be overshadowed by infarred and optical systems but limitations to the latter, in particular their disadvantages in fog, dust, rain, and nighttime viewing support further development of the former. As with many technologies, the number of applications will increase with the passage of time. The typical millimeter microwave integrated circuit contains associated active and passive elements interconnected by transmission lines. In integrating these components together, various discontinuities arise where evanescent fields and surface waves play an important role in their operation. Work has been done to model these discontinuities with lumped elements but the numerical techniques used to derive equivalent circuits are either frequency bound or dependent. Here we present a full-wave analysis, which in not frequency bound or dependent, to analyze an air-bridge structure. The resultant expressions and methods used are general enough to be applied to an array of three dimensional problems. 1

J.2x a One dimnJfonal view highlUgting curfent dirmctons z b. Two dirwnelonl vilw hihtlW1ltng rllmf ewmo / 1 / 4 d_ I I —- g-o -- c.?hme dlrmneii w hVMW ne r Mfttw tOlemnmy Figure 1: Air-bridge in enclosed microetrip. 2 Evaluation of the unknown current In our problem the current distribution on the structure of interest must be determined accurately. Then by the use of an ideal transmission line model, the scattering parameters can be evaluated. In obtaining the current distribution, Pocktnigton's integral equation is solved numerically. The formulation of the Pocktnigton's integral equation and the solution for the given structure are presented in the following work. Our structure under study is shown in Fig.l. 2

2.1 Formulation of the Integral Equation Through the manipulation of Maxwell's equations 1 V x T = -jwRc (1) V xf = j=w AT +J (2) V- eE=p (3) V. f/=0 O(4) along with the representation of the magnetic vector potential A =...(5) one arrives at an expression relating current and the magnetic vector potential V2 + k'= -J7. (6) When the current 7 is represented by a dirac delta function in equation (6), the Green's function becomes a solution as shown by the equation van + k07 = -jA76(7 - r). (7) To obtain a unique solution that applies to the specific geometry as shown in Fig.l, one must apply the characteristic boundary conditions of the structure. Note that we have introduced the dyadic form as we need to be able to describe fields which are produced by a current of arbitrary orientation. 1 throughout this report an eJ time convention is asumed and supprmued 3

For the case of a single x-directed current, the unit dyadic 7 takes on the form; 7 = x. (8) The vector representation of this current is J = 8(7 - T)2. (9) Equation (9) represents a dipole directed in the x-direction and parallel to the interface between regions II and III. It has been shown by Sommerfeld that the magnetic vector potential of this structure needs to have two components so that the appropriate boundary conditions are satisfied. This dictates that A must have one component parallel to the current source and another parallel to the interface; D = A'X + A!i. (10) The integral equation which relates the magnetic vector potential to the current of interest is written as = J Jf.3dv (11) where the dyadic Green's function i is uniquely defined by the structure under investigation and takes on the general form G=xx Gtifxy Gxi = G..y: G.yy GsI5 ai denotes for which region (I, II, or III) the equMioan pplie m denaed in Fig.2 4

In the case of a single directional current in the x-direction, the dyadic equation has only two components, GT = &sxx + zx X2. (12) From equations (1)-(5) our electric field is related to the magnetic vector potential by VxH 1 1 = -=-V x (V x A) = —(k2 + VV 7i) (13) jc~c jWe/ JW4 therefore E =-jw 6 (6A3 6 Az )1 (14) Et[ 1 (A)] (15) E =-jw A +(k)2 A 6Az ) (15) and through Maxwell's equations, He 1 (6 SA H. = ---- (17) Hv)H [I T~ ( Az,) (18) 1 (6A,\ Hs by at (19) 2.2 Derivation of Green's function For our structure pictured in Fig.l (an air bridge on an enclosed thin microstrip), we require only the derivation of the Green's function for an x- and z-directed current. For the variation of current in the y-direction, we have assumed a Maxwellian distribution. Our first step will be to derive the Green's function considering the x and z components of the current separately. 5

- a - — 1\a %i Region 1 I — -, —-, --- -- Region 2 (xIy#= b() T Region 3 h z Figure 2: x-directed current above dielectric in enclosed microstrip structure. 2.2.1 Derivation of the Green's function for a x-directed current With equations (14)-(19) the Green's function can be related to the electric and magnetic fields which have conceivably defined values dictated by the structures electrical characteristics. We begin by stating that the tangential electric fields are zero everywhere on the surface of the walls of the structure; E~ =Oat=0O,a (20) EZ =0 at y = 0,b (21) El, = 0 at z =0 (22) Ezu = 0 at z = c. (23) Employing separation of variables to the expressions and the established boundary conditions (20)-(23), the following general forms of the green's functions can be derived for each region of interest 6

Gs= 2n A'cos (-) sin( (y) sin(kz) (24) G Bin sin ( y) S(kz) (25) Gz = cE o (-) sin (!ry) (A"sin(klz) + Czcos(kfz)) (26) n-O m-= / \ G "= os sin?r (B" sin(Ik"z) + D"cos(k'z)) (27) G'I l l A cos sin i( ) sin (k"(z - c)) (28) ns"Oms \= 1 \ 0 / G' E B'""sin sin sin(k -c)). (29) n=i m=i In equations (24)-(29) the eigenvalues k1, k,, and k, satisfy the following relations: (k')2 = (ki)2 + (k;)2 + (kf)2 where k, = (-) and k, = (m ) k2 = w2/e. (30) To determine the unknown coefficients A1"t"1111 and B111111"l, one should apply boundary conditions at the two interfaces between Regions I, II, and III. Since at 7

the boundary between Regions I and II there exists no magnetic charge, the normal magnetic field must be continuous across the boundary. Therefore H! = H.", so A'sin(kz') = A""sin(k"z') + C"sin(k!'z') (31) Since there is no electric current in the y direction, the tangential magnetic field is continuous. By substituting this relation into (18), one obtains HI = H" therefore B'cos(klz') = B"sin(klz') + DIIco(kIz') (32) Also, since there is no magnetic current in the x direction, the tangential electric field is continuous. Therefore El = E4l; A'sin(kz') - B'sin(kiz') = B"cos(k"z') - Dlsin(k"z') + Alsin(k!z') + C"sin(k'1z') (33) By applying similar boundary conditions on the interface between Region II and III, the following equations result; A"sin(kt'H) + C"cos(kt'H) = A'tsin(kl(H - c)) (34) B"sin(kf'H) + D"cos(k!'H) = Bl"cos(kt"(H - c)) (35) krt(A"coe(kl'H) - C"sin(kt'H)) = All'kll"cos(kl"(H - c)) (36) A"sin.(ktH) + C"cos(kLH) (37) V' + B ) (B"cos(k"IH) - DlDsin(k"H)) =L Atsin(kt"(H - c)) + B" k n(1(H- c))) 8

Since there does exist an electric current between Regions I and II, the magnetic field between these two boundaries is discontinuous. This discontinunity can be considered by integrating the inhomogenous helmholtz expression over the boundary and using orthogonality. This results in ab (kA'cos(kfz ) - kA"cos( z k') + kl'.C"sin(kz')) = —- - A (4b wY |.-.co^ m^s ry (38) ( 4 whenn O0 where p = ( 2 when n=O Using these eight relations derived from the boundary conditions, the eight unknown constants, A', Bl, All, B"t, C"'I, D1", A"11, B11 are found. The resulting Green's functions for a x-directed current above the dielectric upon simplification are oop [it -t"i- t" - cot(kH),o mnl abk,(cot(kz')cos(kz')- sin(k,z')] CME r - in "ry. }3 - --- sin(kc^) (39) =, p(l - tan2(k.H)) () ( t- 1)tan(kab(Ht - c)) a (= 11tan(k, ) - ktan(k11(Ht - c))[cot(k.z')cos(k,z') - sin(kz' )] n=Omrn I Cj int. (my 1 COB -a ) k b ) kitan,(kl(H - c)) - ~lIk,tan(k,H) 9

sin( ) sin b cos (kz) (40) ~ co p [kflr - k-tan(kI"(H - c))tan(k,H)I n=O mn abk[(cot(kz')cos(kz') -sin(k,z')][klltan(k,H) - ktan(kl(H - c))] fnrx\. mn y n \. (mbry\ cos -) sin ) sn(k- z)cos ( sIn m -- y + a b a b abkcot(kz')c(kz') C — in(k S ( ) sin (m ) cos(k z) abk,[cot(k.z')cos(kz') - sin(k.z') a bs (nrx \. (miry cos )sn b (41) 0 0 p (1 tan2(k.H)) (=) ("e - l)tan(k."(H - c)) nOm ab[kl"ltan(k.H) - ktan(ki"(H - c))][cot(kz)co(kz') - in(k,z')] __1____. I nirx \ ( mvry\) Cos a )Mn byr (42) CW = fp (1 -tan2(k,H))cos(kH) n=Ol ab[cot(k.z')co(k.z') - sin(kz')lcos(kl(H - c)) [k"ltan(k.H) - ktan(kll(H - c))] ~ni ~nrx\i mry (k' - nvr\ (miry cos ir n -\ n(k(z c))cos nr- sI M( - (43) = a p(l - tan2(kH)) (f) (l - 1)tan(kl"(H - c)) nm ablkllltan(k,H) - k,tan(klll(H - c))][cot(k,z')cos(kz') - sin(k z')] - sec(kll(H - c)) [kflltan(k,(H - c)) - e"tan(klt"H)] sin (-) s ( ) cos(k(z - c))cos - sin (( - )) (44) 10

Region 1 Y (x'.y.z H Region 3 z Figure 3: Structure when conducting strip i lowered on to the dielectric. If the conducting strip is lowered on to the dielectric, we will have only two regions; air representing Region I and dielectric representing Region III. To formulate the following Green's fuctions for the strip on the dielectric, one simply lets z' = H in the previous expressions. = pA. Ik ) p tan(kll(H - c)) Cz n=m l ab(cos(kH))[kIltan(kH) - k.tan(kIII(H- c)) (nirx \." ( (n rx\. (rrwy\. c cos s sn szn (kz) (45) a/ \ b f \ a / \ b / 0_ - 0 p tan(k,H) (v) (e4" - 1)tan(k"f(H - c)); nO ~.= ab(cos(kH))(ki"tan(k,H) - ktan(k:"(H - c))1 11

_1 _____ (n \. (mTry\ (k:"tan(kt1f(H - c)) -elCktan(k*H)J a ) S b) sin ( ) sn (b y)co(kfz) (46) CG11 -l f'f-__________p tan(k,H) Xs nO ml abcos(klI"(H - ))(k;:"t(kH) kta(k"(H - c))] (nrx\ (mlry\ (nirx\.(mry \ cos a-) s ( s) n(k"(z - c))cos -) sin (r (47) COS a) kb / ~n~k~' ~ZCJJCOS a b = ~ X p(1f" - )tan(kf"(H - c)tan(kH) (*) I: =mlm abcos(k,(H - c))[k"tan(kH) - k.atan(k"(H - c))] ___1________ ( fnrx\ fmry\ 1 sin 3i( cos~k1I ( - c)) [ki"tan(ki"(H - c)) - el1ktan(kXH)] n ) c(k(z c)) COS (- ),s( b (48) 2.2.2 Derivation of the Green's function for a s-directed current. The structure used to consider the z-directed current is pictured in Fig.4. We will simplify our problem separating it into two parts; a primary field problem and a secondary field problem as shown in Fig.5. For a z-directed current only one component of the magnetic vector potential is needed to satisfy the boundary conditions; A:= iAz (49) 12

X % SI Region 3..... IIl I Ix~ a! Reggion 2 I Reg ^0q^Region w^ 3 I Zs) i z Ye' H 0 l H E(x.YaZ.) H Figure 4: A z-directed current above a dielectric substrate. Region1 Region 1 Region 1 Region 2 || = Region 2 | + Region 2 egion3 h Region 3 h z z z Fgr 2 i Figure 5: The problem is diveided into two parts, a primary and secondary problem. 13 [3

where Af = Al + Al, (50) Al = Afl + Af; (51) Al" = All. (52) From our definition of Magnetic vector potentials, VxA 7= (53) and from Maxwell's equations, Vx _ 1 1 =.-v x (V x ) - (k2A +7/A) (54) therefore, for the z-directed current, Ex=j (k + ) A, (55) w = (6,\A ) (56) jwC \6ybz E,, = -we 6''z (57) H, =O (58) f (-ZA,) (59) H by/A,) (60) Now we must apply the boundary conditions of the structure to obtain the general (primary) solution. The first readily known boundary conditions are those on the walls where the tangential electric fields become zero. Therefore E,'=Oatz=Oy=O,b (61) 14

EI = 0 aty = O,b (62) E = 0atz=; =,a (63) E O at x = O,a (64) E =0 at x,ay=O,b (65) El =0 at = O,a =O,b (66) In the primary field problem, the inhomogenous differential equation takes on the form V2A, + k2A, = -p7. (67) The solution to the above equation when the listed boundary conditions are satisfied is of the form Al E E Alsin ( - sin (m y co(ktz) (68) iiml mini RiminAsp a / All = E E A0 m (-) si ( ) e-W- (69) In the secondary field problem, we will derive a solution that satisfies the homogenous differential equation. This is due to the fact that we do not have a current source in the secondary field problem. In both field problems, the electric fields must satisfy the same boundary conditions on the conducting walls. As a result, the secondary fields are of the same dependence with respect to the x and y coordinates, therefore 15

AI v^ fif 4 nrx \' mwy n=el m= j abk, a b A = (a ) Sin(a) sin' (in Irx-sin( bY )cos(kCz) (70) All = E(f."sin(k.z) + flcos(kz)) (j abk nl m=l \ abk (1rx n (myry srn -a-) sin \ cos(ksz) i ) i (b ) (k' 3m(-)s39 mbry) (71) For Region III we have the same standing wave solution as in Region 1 except for the fact that the conducting wall has been moved by (H+h) along the z-axis as reflected in our z dependence below Aln= - >j 4P j ik,,e fItsin n'z sin m(ry' s1 ~ a IIIb sin ( ) sin ( y) cos(^k"(z -(H + h)). (72) In the case of a delta function source, A,,sp will give a G,,p component in the dyadic Green's function and similarly, a A,. will give a G,,,. These two components are related to G,, by the relation Gs = Gsp + GC,, (73) Since we have a current source between Regions I and II, our magnetic field is discontinuous so we must integrate the inhomogenous helmholtz equation over the 16

interface which results in lim e + (V2 + k2)A,dz = -+ lim o 6(z - xz)6(y - y')6(z - z)dz, (74) a-Oru' "+ -x'O #-' -74 and upon simplification, one obtains lim ( A. +a = -6(z - x')6(y - y) (75) 3-o. A m I,_= = Ai' I'- =,62- )6(y - y)) (76) from (25), one obtains a (k'sin(k.z')A' - jk"'A) = sin ) n ( ). (77) One more equation is needed to solve for the two unknowns A;, and At'. Utilizing the fact that the E field in the z- direction must be continuous at the boundary between regions I and II; one obtains A'cos(kz') = AI. (78) These equations are solved for the unknown coefficients resulting in A# nirz' \ y A = bk, ejk sin (- )sin (79) j4(abka sin (a) sin (2 ) cos(kz'). (80) By substituting these expressions into our general forms (68) and (69), the zzcomponent of the Green's function takes the form; Gl4= en k 4a e J sin nz sin My sin l n a y —) cos(k,) (81) 17

n=l m=1 abk, abJ s in- s in ) (82) In order to determine the four unknowns of the secondary field problem, boundary conditions on the air-dielectric must be applied to the structure of Fig.5c. First, the z-component of the electric field is continuous across the boundary between regions I and II; El = ElI. Therefore f'cos(kz') - f'sin(kz') - f11cos(kz) = 0 (83) Integrating the homogenous helmholtz equation (there is no electric source in the secondary problem) across the boundary and using orthogonality, one obtains fIcos(k,z') - flsin(k,z') + fIsin(kz') = 0. (84) From (83) and (84), we conclude that f' = (85) and fr = L". (86) From the boundary conditions at the dielectric interface, E[1 = Ell one obtains je-JklHco(kz') + ft in(kH) = -f sn(k h) (87) -r rr and from H,' = H"', e-it'Hcos(k,z') + f"cos(kH) 1 fIcos(kIl^h) I=, —— "' k1" (88) 18

The resultant Green's function for this problem will have a zz component only, which is given by Z lm=11 +f )jabk, a )sSTbI sin (-) sin ( b 7) cos(kJz) (89) G"- j L 4, eik.'(ei*s.cos(k.z) + f fsin(k,z) + fccos(k.z)) ci~ =n~m1 abk, ngl mmli (n9 \. (mn y\ /a ) b ssi sin( r) (90) f"Li nirz mlry G~ = E 3 a ) sin G'" =il1 mml ab ) S?7L b))i sin (-) sin( 2Y cos(k"'(z _ (H + h))). (91) After applying all the necessary boundary conditions, the Green's function takes on the following form l OC f 4"Jk,cos(k.(z' - H))cos(kflh) + kf4sin(k,(z' - H))sin(kfl"h) =~ kIkejtsin(kH)cos(kI"h) + ki"k,sin(kl"h)cos(k.H) 4__ f. ix. nmy' in -- sin mryn ab ( a ) ( b ) min ( —j sin( lryb CO(k?,z) (92) 19

l e"''"kcos(k,(z - H))cos(k^ "h) + kif" sin(k,(z - H))sin(kh) nlm l k24Zf1sin(k.H)cos(kJ'1h) + krt"ksin(k,"h)cos(k,H)'sb a i b bsin ( a ) sin (b cs(93) __m ~f00 00 4e1_"kj"cos(k.z') 2Z n=lm= k2 ci sin(kH)cos(k "h) + kII sin(kI"h)cos(kH) 41. nrz'. mry -Lin ( sin ( - b cos(k z ) sinin (f Y) cos(k1(z - (h+ H))) (94) 2.2.3 Summary of Green's function determination In this chapter we have determined the unique Green's function for the airbridge structure (Fig. 1). This was accomplished by working with Maxwell's equations to establish a tractable equation and by representation of our source as dirac delta functions. We then applied boundary conditions to solve for the unknown coefficients and formulated a solution. We now have expressions that will give the resulting field produced by a point source directed in the x or z direction as required to analyze the air-bridge structure. 2.3 Application of Method of Moments The method of moments is a numerical technique used for solving functional equations which cannot be solved in closed form. By reducing the functional relation to a matrix equation, known techniques can be used to solve the resulting 20

matrix equation. This method is computationally intensive but with the advent of faster computers, the method has become feasible. To apply the method of moments in the specific case of an air-bridge, one should follow the steps outlined below: 1. Use the integral equation (11) derived in section 1 along with the relations (13) and (5) so one obtains an integral equation that relates the current to the electric and magnetic fields respectively. A general form of this is Lo(J.) = S (95) where Lo, is an integral operator operating along with the derived Green's function and 7 is a vector function of either the electric field TE or magnetic field H. 2. Represent the current on the conducting strip as a sum of coefficients multiplied by a pre-determined basis function, Nia, = E I97 (96) qmi In equation (96) where I, represents the complex coefficients, Nia represents the number of sections the strip is to be divided into and 7q represents the chosen basis functions which represent the current distribution. 3. Discretize the integral equation by minimizing the resuting error function 6E on the surface of the conducting strips. In applying the above steps to the problem of an air-bridge we have set up the integral equation for the electric field of the form r= - i/w (7+ vv)..7dV. (97) 21

J 2x J. O di i i i i i c i i2z-J3xon a. One dimensional view highlighting current directions Figure 6: Currents are assigned variables by direction. In proceeding to the second step, we seperate the problem into five different sections as pictured below (this figure is taken in part from fig.l). For J1s and J3, we will model the current as a sum of an incident current A, a reflected current B, and the sum of incremental currents Iq. For Ji2, J15, and J2, only a sum of incremental currents is required. Implementing this convention results in NIC JS = Ale" + B.e-," + E IqlJ7 (98) q=1 N23 J2, = E Iq2s7 (99) e lq==J1 232 J3/ = A,.," + Bu~e,-" + C 7,.7,, (100) q=l 22

NiX JA. = E IqlJl7z (101) J2 = E Ig,2J2zq. (102) The basis functions the x-directed currents in this case are the same and are defined as being! in(lI,) zqz -1- S * z.,.(.)(J^ z,-1 — < _< < x+ =isg~ BYZ) l F Zn(* q Z' < Zq+ (103),(-),X T ( Y 10) where for the y-direction we hae used a maxwellian distribution function. The first step in discretizing the integral equation (97) is to evaluate the error that our mathematical representation of the electric field will produce. This is done by evaluating the electric field produced by one section of current on the conductor on another section of the conductor. Since the electric field on any part of the strip must be zero, the value of our integral evaluates the error. Then, by the concept of least square estimation, when one takes the inner product of the basis function along with the error and sets the result to zero, the error is minimized. Proceeding to do this for our problem here we first account for the different components of the fields produced by a current of given orientation. E, = E,1 + E2, + E~, + E1,, + E2,s (104) 23

and E, = E1,, + Eu, + E3,x + E1,, + E2,,. (105) First, we shall consider the x-directed field produced by a x-directed current on the dielectric El: = Aix C (-Z ) e dz' if ( -) 1 (2b It) dy *r ___, p tan(k;II(H - c)) 1ab(cos(k.H))[k.I"tan(kH) - ktan(k."(H- c))] 2 ( )co (-) in (r ) in(kz) + p tan(kH) () (e -1)tan(kl(H - c)) abcos(k.H)[kI"Utan(kH) - k tan(k.U"(H - c))] 7t r cos -n- )s m cos(k2)] 4 [km"tan(k?"(H - c)) - (")k,tan(k,H)] a cos R1iss | cos ( S dz' (-) ()Iin ( b y( ) dy' a\ VW 2Ir [ __ PtP tan(kll(H - c)) ab(cw(k,H))l[k1tan(k,H) - k,tan(kII(H c))] (n) COJ ) sn (b ) sin(k.z) + 24

p tan(k.H) (=) (ct" - 1)tan(k;"(H - c)) abcos(kH)[kf,)tan(nkH) - k,tan(k(Ill(H - c))] 1 fn mx\ [kX"tan(ki"(H - c)) - e"llk,tan(kH)c a ) san b) COS(kz) + NI sqi I(f sin(k( 1) ) s d + 1 sin(kn(kl l,) ) a [(sk2 in) p tan(ks(H - c))+ aco l -) sin ( - S3in(fz) + d p tan(kH) ()( - )tan(k"(- c)) abcos(k,H)[k"tan(k.HH) - k.tan(kll(H - c))] [ki"tan(kil"(H - c)) - ektan(k H)] (-) k,cos (-) sin (% y) cos(kz) (106) 25

th~ xr nCnionm; of thr rluotria finArlodn r-nit ogM ia —l For a x-directed current, a z-directed field is also produced. Proceeding with the same procedure as above one obtains EIz = AI cos -1: eik dX in ( b dy' f ____r _ p tan(k,"(H - c)) [ab(coa(ikH))([Ik"tan(k;,H) - k,tan(ki"(ff - c))] (22jr)n (Z) kcoa(1,z) + p tan(kH) () (e1 - l)tan(kH"( - c)) abcos(kH)([kitan(k,H) - k,tan(klf(H - c))] [kiI"tan(kI"1(H - c)) - e4Ik.tan(kH)] (k2 - k;2)sin m- sin 1 y )sin(k)] + f nr I f 2 \ I n RI / cot (- { -z | I -) 4 S sin ( b Y ) dy a-f \I /-f'rw 2 - 3Y f __ ___ p tan(k1"(H - c)) [ab(cos(k,H))[tkItan(kH) - k,tan(kl(H - c))] (-)sn ( ) sin ( Y) kcos(kz) + 26

p tan(k,H) (Z) (e - 1)tan(k(ff(H - c)) abcos(kH)[k.mtan(k.,H) - ktan(kl"(H - c))l [kll"tan(kl"l(H - c)) - ellk,tan(k,H)l (k2 - k2)sin (- sin ( b y)ain(kz)] + zVm\ a \ E- 1'-: sin(k(.' z-..)) ) + qo nl sin(k),) a s in(k(z+i - XI)) m Jo sin(kl.) a /2 1. mir A, L (-h) -- ()-r]s (7 v') dy* ___r p tan(kIl(H - c)) ab(cos(k,H))[kx"Itan(kH) - ktan(kztz(H - c))] /-nx\r. /n~~x. /w ry\. v (-) sin (-)Sin b kocoa(ksz) + p tar(kbyH) (A) (41 - 1)tan(kl"(H - c)) abc,(kH)[k1tcman(kH) - ktan(k?"(H - c))] [k"ltan(ki"(H - c)) - e"ktan(k.H)] 27

(k2,'- k)Sin ( ) sin (by) sin(k,z)] (107) For a z-directed current, one must consider the fields above and below the source seprately as they have different forms in those two respective regions. Also, as before, both x- and z-components of the electric field are produced. First we consider the x-directed field produced by a z-directed current above the source where Ni, = S ^ ( ^sin(k(z' - zq)i)) El,,. -- = q1, ( sin(k(zl) )("~Ik.,cos(k,(z' - H))cos(klllh)+ k'"sin(k,(z - H))sin(kt"h)dz) + /'f sin,(k()s+ - z'))(4"kco.(k.(; - H))cos(k.m'h) + JO sin(kl,) k"'sin(k,(z' - H)).in(k"h)dz)) /d t 2 \ 1. mr.dy. -![1-(i) ] 4p1. n \ ab etk"sin(kH)cos(k Ih) + ktkcos(kH)sin(kih) in a ) (-) cos -- sin( -- )(-k,)sin(kz) (108) For the xdirected field below the source produced by a z-directed current, one 28

obtains Ni s E sin(k(z -*-',/)),,- sin(k(z+l - z')) 4p eIksin(k(z - Hcos(k _h) + k"Icos(k,(z - ))in(k"). /n, ab k,(e.sin( kH)co)s(k("h) + ksin(kh)cos( ) n V (-) os (-)^ C 3( fr y) (109) For the z-directed field above the source produced by a z-directed current, one obtains V( sin(k(z' - zg.d)) Iv= (i s1n(fk-i-'.l()"k.(4"kcos(k(z' -H))cos(k"h)+ ksini(k.(z' - H))si((kIh)dz') + / sn(k ) ((k*(z coj(i(kZ - H))cos(kzl-h) + pr \ 2 inskin) ak,in(k,( H - H))ein(kIlIh)dz')) /L (t —) 1i (ib y') dy' 29

4_ 1. n) + ab i"k, sin(k, -H)cos(ki"h) + k"tkcos(k, H)sin(kI"Ih) a ab2 sirk\ rz\ I fmry\ (k2 - k2)sin (-) -sin - - ) cos(k,z) (110) For the z-directed field below the source produced by a z-directed current, one obtains El, = Iq ( n(k(z L cos(kz )dz + cos((kz') dz"' q1 = sin(<k,) Jo sin( kl,) 41. MnTr, VW 21 sin (') dy' sin I - 2 ab ( ) / Ce1l1ksin((z - H))cos(kflh) + klcos(k,(z - H))sin(kzirh) - k,e4"lsin(kH)cos(k"Jbh) + kflIsin(kIl"h)cos(k.H) (k - k, ^ - n2)s4in )rY (111 For a x-directed field in the region above a x-directed current which is above the dielectric, one obtains?e =^, / sin(k(zx - zl)) /n E1 -s~ OM1 f 3 — in(kl,-) a- f sin(k(z+i - z')) / n Jo sin(kl) \ a ) ) 30

/t (/ 2 ) 1 in ( my') dy' r (-) si (- -ktain(k -ct) +H (k ( L:p [ 1"tar~k.H)-I.,1(;f-c)) — cot( k,H')] [ abk. [cot(k z')cos(k,z') - sin(k.z')] c ~bos m s-i m sin(k,z) + p(l - tan2(kH)) () (eI - l)tan(kf"(H- c)) ab[k(l.mtan(kH) - k.tan(k" (H - c))][cot(kz')cos(kz') - sin(k,z')] klltan(klt"(H - c)) - elt"ktan(kH) (-) k,cos (-) sin mry) (-ksin(k,z)] (112) For the x-directed field located below the x-directed current source which is above the dielectric, we obtain PElr =, ( sin(k(z' - 21))x (r dz'+ f,'n s(k (kl,) a / / cos ( 2 )x sin(kl,) ao L (12) [ 1 sin (~y ),' dy' k - a p [fL" - k.tan(k'"(H - c))tan(kH)\ a bk.[COt(k.:Z')cos(k.z') - si(k:,' )[lJk"tan(kH) - k.tan(k.( -H 31

/. a?\ n.s /myry.,\ _ os ( ) sn (b) ain(klz) + -P / nvx\. miry, ~ abk[cot(kz')cos(kz') - in(kz')] ( a i (i b) cos(k z) + p(l - tan2(kH)) (a) (4" - 1)tan(k"(H - c)) ab(kliItan(kH) - kxtan(kIIf(H- c))][cot(kz')cos(kCz') - sin(k,z')1 kI"tan(kt"(H - c)) - e"Iktan(k.H) (-)kco. n-) sdin ('r) (-k.)sin(Ikz)] For a z-directed field above the current produced from a x-directed current above the dielectric, one obtains the expression N.. in(k(x - Z,.-i)) (nir \ El.. = = I.u!Cos Z sdin E1). = Iq2(i(.m)n(kl,) c a I n(k(z,+i - X)) (n \ o 3in((kl) \ a ) ) I ( 2 s 1 in (b rs ) dy' [i/' (f)- i] nt)I sin ($)P l[. kt-k. -,' _S cot (k.H)] a ) \a abk.[cot(kz')cos(kz') - sin(kz') 32

sin ( bY) (k,)cos(kz) + p(l - tan2(kH)) () (4" - l)tan(kll(H - c)) ab([kIItan(kxH) - k,tan(kkII(H- c))]kcot(k:z')cos(kcz') - sin(k,z')] 1 kII"tan(kII(H - c)) - ell"ktan(kH) (k' - k.)sin ( —) sin (miy cos(k,z)] (114) For a z-directed field below the current produced from a x-directed current above the dielectric, one obtains the expression sin( ( + s -in(k(z) -'1)) (d r ) [in((, -s(k ) si(kz ) tn(kn( H - in( ) (kls) \ a + It 2 1rrT I \ ~D sin - {'y dy' [ _________ ft [~kfJ - k.tan(k1,1( - c))tan(k.H)} abkl[cot(kz')cos(k,z') - sin(k,z')j ([ ta sin ( kH ) - tn((-k)sin(kl) 33

p(l - tan2(kH)) (=) (e" - )tan(k"(H - c)) ab[lkt'tan(kH) - ktan(kH "(H - c))][cot(kz')cos(kz') - sin(kz')1 1 klltan(kl(H - c)) - d4ktan(kH) (k - k2)sin (-j ) sin (tby cos(k+z)] 2.4 Matrix Equation The resulting matrix equation is formed and upon inversion the unknowns can be obtained for a given exciation. 3 Scattering Parameters Using the derived current distribution on the conductors, one can apply an ideal transmission line model to determine the scattering parameters. 4 Summary From this general analysis of the Green's function and the ultimate determination of the scattering parameters of an air-bridge, the formulation used here could be applied to a variety of structures whose geometry requires a three-dimentional analysis. Other planned work includes the application of this work to other structures and the use of air-bridges as a curcuit element in the construction of other passive microwave circuits. 34

References [1] A. Sommerfeld, Partial Differential Equations, New York:Academic Press Inc., 1949. [2] R. Harrington, Field Computation by Moment Methods, New York:The Macmillan Company, 1968. [3J L. P. Dunleavy and P. B. Katehi, Preliminary results for study of theoritical and experimental characherization of discontinuities in shielded microstirip, radiation lab report RL 846, August 1987. [4] R. Harrington, Time-Harmonic Electromagnetic Fields, New York:McGrawHill, 1961. [5] T. Hirota, Y. Tarusawa, H. Ogawa, "Uniplanar MMIC Hybrids-A Proposed New MMIC Structure" IEEE Transactions on Microwave Theory and Tecniques, Vol. MTT-S5, No.6, June 1987 pp. 576-581. [6] R. E. Collin, Field Theory of Guided Waves, McGraw-Hill, New York, 1960 35