A study of microstrip transmission line parameters utilizing image theory

Transcription

1 Sholars' Mine Masters Theses Student Researh & Creative Works 1970 A study of mirostrip transmission line parameters utilizing image theory Joseph Louis Van Meter Follow this and additional works at: Part of the Eletrial and Computer Engineering Commons Department: Eletrial and Computer Engineering Reommended Citation Van Meter, Joseph Louis, "A study of mirostrip transmission line parameters utilizing image theory" (1970). Masters Theses This Thesis - Open Aess is brought to you for free and open aess by Sholars' Mine. It has been aepted for inlusion in Masters Theses by an authorized administrator of Sholars' Mine. This work is proteted by U. S. Copyright Law. Unauthorized use inluding reprodution for redistribution requires the permission of the opyright holder. For more information, please ontat sholarsmine@mst.edu.

2 A STUDY OF MICROSTRIP TRANSMISSION LINE PARAMETERS UTILIZING IMAGE THEORY BY JOSEPH LOUIS VAN METER, A THESIS Presented to the Faulty of the Graduate Shool of the UNIVERSITY OF MISSOURI - ROLLA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE IN E~ECTRICAL ENGINEERING 1971 T563.!J~ ~d~4' 3 pages ~ (Advisor) (/ ~944?

3 ii ABSTRACT This paper is a theoretial investigation of the potential, eletri field, apaitane, and harateristi impedane of the open strip transmission line or mirostrip onfiguration based upon the lassial Thomson Image Tehnique. It provides the basis for determination of the harge distribution on the strip and reports impedane values whih ompare favorably both with experimental values and theoretial work in the urrent literature.

4 iii ACKNOWLEDGEMENT The author wishes to sinerely aknowledge the assistane and guidane given him by Dr. James Adair. His readiness to provide time for helpful disussion, suggestions and omments will always be remembered.

5 iv TABLE OF CONTENTS Page ABSTRACT ACKNOWLEDGEMENT LIST OF ILLUSTRATIONS LIST OF SYMBOLS I. INTRODUCTION II. HISTORICAL REVIEW A. Literature Review III. THEORY B. State of the Art. Objet of Investigation A. Determination of the Eletri Field and Potential by the Image Tehnique 1. The Line Charge in Front of a Ground Plane. The Potential for a Line Charge in Front of a Dieletri Slab 3. Appliation of the Image Tehnique to the Mirostrip for Determination of the Potential and Eletri Field a. Determination of the Charge Configuration b. Determination of the Values of p and p " L L. The Eletri Field of the Filamentary Charge in the Region Above the Ground Plane ii iii vi viii

6 v Page d. Determination of the Eletri Field of the Upper Condutor in the Region Above the Ground Plane 5 e. Determination of the Potential at a Point Above the Ground Plane due to the Conduting Strip 39 B. Expressions for Capaitane and Impedane A First Approximation to the Capaitane 45. Higher Order Approximations to the Capaitane The Impedane Problem 53 IV. CONCLUSIONS, DISCUSSION AND SUGGESTIONS FOR FURTHER WORK A. Conlusions and Disussion B. Suggestions for Further Work VITA 63 BIBLIOGRAPHY 64

7 vi LIST OF ILLUSTRATIONS Page Filamentary Line Charge Above an Infinite Ground Plane Profile View of the Ground Plane and Line Charge With the Image Line Charge in Plae Filamentary Line Charge in Front of a Semi-Infinite Dieletri Slab Profile View of the Dieletri, Interfae, and Line Charge With the Image Line Charge in Plae A Profile View of the Mirostrip Showing Point P Whih Represents an Axial Filamentary Charge on the Surfae of the Upper Condutor The First Stage of the Image Solution Showing the Fititious Dieletri and the Image of pl in the Air-Dieletri Boundary Illustration of the Addition of the Image in the Ground Plane of the Original Line Charge pl The Final Charge Configuration With the Third Image Charge Plaed at Y = 3a + b The Charge Configuration for the Potential Inside the Dieletri Illustration of the Primed Coordinates and the Dimensions and Boundaries of the Upper Condutor Mirostrip Capaitane Versus Condutor Width-to-Dieletri Thikness Ratio (For ~/a values 1-15 and ~r values 1-10)

8 vii Figure Page Mirostrip Capaitane Versus Condutor Width-to-Dieletri Thikness Ratio (For ~/a values 1-15 and r values 10-90) Mirostrip Capaitane Versus Condutor Width-to-Dieletri Thikness Ratio (For ~/a values and r values 1-10) Mirostrip Capaitane Versus Condutor Width-to-Dieletri Thikness Ratio (For ~/a values and values 10-90) r A Comparison of Raw Theoretial and Experimental Impedane Values Theoretial and Experimental Impedane Plots (a = 1/16") Theoretial and Experimental Impedane Plots (a = 1/3")

9 viii LIST OF SYMBOLS A a a X a y B b pl/ti 0 The Thikness of the Mirostrip Dieletri A Unit Vetor in the X Diretion A Unit Vetor in the Y Diretion pl/ti 0(l + r) The Height of the Filamentary Charge Above the Air-Dieletri Interfae Capaitane An Eletri Field Vetor A Total Eletri Field Vetor The Eletri Field in Region 1 The Eletri Field in Region The Mirostrip Eletri Field Contributed by the Bottom Surfae of the Upper Conduting Strip The Mirostrip Eletri Field Contributed by the Right Vertial Surfae of the Upper Conduting Strip The Mirostrip Eletri Field Contributed by the Upper Surfae of the Upper Conduting Strip The Mirostrip Eletri Field Contributed by the Left Vertial Surfae of the Upper Conduting Strip The Width of the Upper Condutor of the Mirostrip p Q r An Arbitrary Point of Interest at Whih a Potential or Eletri Field Value is Being Determined The Charge Residing on the Upper Condutor of the Miros trip The Distane From Point P to pl

10 ix r' The Distane From Point P to the Image Line Charge The Thikness of the Upper Condutor of the Miros trip X y z z z r The Width Coordinate of the Mirostrip The Height Coordinate on the Mirostrip The Length Coordinate on the Mirostrip Wave Impedane (Bar Distinguishes Between Impedane and Z Coordinate) Charateristi Impedane of the Mirostripline (The Phrase "The Impedane" Unless Otherwise Stated Refers to the Charateristi Impedane) (The Bar Distinguishes from the Z Length Coordinate) Propagation Constant An Arbitrary Dieletri Constant -9 The Permittivity of Free Spae - l/36rr x 10 Farad/ Meter E: r The Relative Permittivity of an Arbitrary Material The Permittivity in Region 1 s The Permittivity in Region A Line of Filamentary Charge p L I An Image Filamentary Charge Resulting From the Air Dieletri Boundary p L II An Image Filamentary Charge Resulting From the Ground Plane Potential Funtion Potential Due to a Filament of Charge Potential Due to the Total Conduting Strip Potential Funtion in Region 1 Potential Funtion in Region 00 Infinity

11 1 I. INTRODUCTION Reent work in the field of semiondutor physis has brought to reality a suitable line of solid state devies whih operate in the gigahertz range. To omplement this ever growing group of ative devies, the designer has looked to the stripline and mirostrip onfigurations as a passive interonnetion, thus srapping the old-fashioned waveguide plumbing and oaxial onnetions for more onvenient strutures. In addition, the mirostrip holds some promise to the omputer designer, whose objetive is to minimize the transmission times of impulses traveling between interonneted devies. Fortunately, the study of mirominiaturization tehniques, whih is neessary for the thin and thik film tehnologies, has also reently been aelerated. This makes it possible to exert a high degree of ontrol on the geometries of mirominiature devies like the mirostripline, whose harateristis vary so greatly with geometry, size, and purity of material. With the realization of suh advanes must ome a sound theoretial knowledge of these devies. This paper is a study of the mirostripline and its parameters based upon sound theoretial priniples.

12 II. HISTORICAL REVIEW A. Literature Review The mirostrip transmission line and its history are often onfused with that of the symmetrial or losed stripline, but distint evidene of work in the area an 1 be found as far bak as the 1930's. However, the idea remained reasonably obsure until the early 1950's ' 3 ' 4, when work in the gigahertz region was being expanded. In an attempt to introdue the user to the various stripline onfigurations, the IRE prepared a speial "Symposium on Mirowave Strip Ciruits" in 1955 whih inluded some papers ataloging what was then the state of the art in mirostrip theory5 ' 6. In their analysis, Blak and Higgins' attempt to use the Shwartz-Christoffel transformation was not entirely suessful beause of their inability to solve some of the key equations in the overall solution. Beause of the lak of symmetry in the mirostrip goemetry and beause of the apparent mathematial problems involved in rigorous solutions, some investigators 7 turned to analog models. wu8 realized that part of the problem was that the mode was not TEM. Aordingly, he solved the problem by starting from urrent equations, assuming that the transverse urrent omponent was not neessarily zero, and

13 3 made approximations for a speial ase and its solution. In spite of this knowledge, most if not all of the subsequent investigators have assumed a TEM mode in their mathematial analyses. Reently, various approahes have been made to irumvent the surprisingly ompliated mirostrip impedane problem. 9 Kaupp approahed the problem by merely assuming a lossless wire-over-ground transmission line using standard TEM transmission line theory and a geometry equivalene to produe reasonably aurate results Wheeler ' approahed the problem with a novel use of the Shwartz-Christoffel transformation from whih he produed very good results. In fat, due to the sarity of good experimental data, most subsequent theoretiians ompare their results with Wheeler rather than experimental values. Stinehelfer 1 utilized finite differenes with relaxation to solve the boundary value differential equations for the mirostrip geometry in a numerial alulation of the potential and impedane.. 13,14 d. t' 1 t h. f Yarnash1ta use a var1a 1ona e n1que or the solution of the impedane problem. Suspet in the latter work is the assumption that the harge distribution should take on that of a thin ondutor in free spae15 with no other materials in its proximity.

14 4 One of the latest attaks on the mirostrip impedane 16 problem was made by Farrer and Adams, who used the method of moments 17 to asertain the potential distribution. B. State of the Art Beause of ontinuing interest, work is moving ever forward in the broad mirostrip researh area. Roughly three sub-areas of researh interest an be defined. These are: (1) Appliation of mirostrips or mirostriplike onfigurations as devies, () Appliation of mirostrips as an inherent part of a total system, and (3) Further researh on the basi mirostrip parameters. Devies whih an be inserted in the iruit by merely hanging the strip geometry have inspired investigators by their simpliity in onstrution. For example, oupled pairs of mirostriplines produe the effet of diretional oupling, bends hange the VSWR, and tapers produe the equivalent of a transformer in a iruit. Many more equivalents are beoming available. In view of their desirability, these are now being studied in d t e a~ '118,19,0 On the other hand, a seond group of investigators has felt that the state of the art has progressed to suh an extent that solid state omponents and mirostriplines ould be mated to produe ompletely operational iruits with pratial funtions. d. h 1' 1,,3 reently been note ~n t e ~terature Several of these have

15 5 A third group of investigators believes that the basi struture and its parameters need more investigation. Some are merely applying new tehniques to the impedane 16 4 problem '. Others are investigating the effets of suh hanges as substitution of ferrites or semiondutors in plae of the dieletri as the base material5, 6. The latter works are revealing that devies suh as mirostrip irulators will be available to miniaturization onsious mirowave engineers in the near future.. Objet of Investigation The objetive of this investigation is tq determine the eletrial parameters of the mirostrip utilizing the image tehnique. It inludes an initial determination of the eletri field and the potential funtion, then a determination of the apaitane and impedane of the line utilizing a first approximation to solve the problem with no initial knowledge of the harge distribution aross the strip. With the potential funtion established it is shown to be possible to determine the harge distribution for a realulation of the mirostrip parameters for higher order approximations.

16 6 III. THEORY A. Determination of the Eletri Field and Potential ~ the Image TehnTque- 1. The Line Charge in Front of a Ground Plane Frequently, the onept of potential determination through the image tehnique is introdued by iting the lassial problem of a line harge in front of an infinite ground plane. Figure 1 illustrates this proposition. The line harge has a uniform harge distribution of pl oulombs per meter, is of infinitesimal diameter and is infinite in length. The ground plane is of infinitesimal thikness, infinite in size and has an infinite ondutivity. The intervening spae is filled with a homogeneous dieletri extending to infinity in all diretions. Not apparent in this problem is an indued harge whih is distributed on the top of the ground plane and is a diret onsequene of the original line harge pl. This also ontributes to the overall eletri field and potential above the ground plane. Thomson7 onsidered this problem and theorized that one or more fititious harges ould be plaed on the lower side of the plane. These so-alled image

17 7 y x~z HOMOGENEOUS MEDIUM - e: Figure 1. Filamentary Line Charge Above an Infinite Ground Plane ---- IMAGE CHARGE -pl r' - IMAGE PLANE -- p \ ---- \ \r \ \ \PL yl ORIGINAL CHARGE X Figure. Profile View of the Ground Plane and Line Charge With the Image Line Charge in Plae

18 8 harges would thus produe the same eletri field and potential as that whih would have been produed by the indued harges residing on the surfae of the image plane. The plane ould then be removed and the problem would be redued to that of a number of finite harges in spae. Thomson's postulate may be utilized in this problem and in any problem where the following onstraints are met: 1. The potential ~ must satisfy the Laplae equation v ~ = 0 exept at the loation of the original harge or harges.. ~ = 0 over the plane Y = ~ = 0 at r ~ = pl/~ 1 ln r as r ~ 0, that is, as P ~ Y 1. For the onfiguration illustrated in Figures 1 and, the funtion ~ = pl/n ln! - pl/~ ln ;, whih is merely the potential of the two line harges, satisfies all four onditions and is thus a solution of the problem. By the uniqueness theorem it is the only solution in the Y > 0 region. In the region Y < 0, ~ = 0 everywhere.. The Potential for a Line Charge in Front of a Dieletri Slab The problem of a line harge plaed in front of an infinite dieletri slab may also be treated by using

19 9 the image tehnique. This proposition is illustrated in Figures 3 and 4. Point P is an arbitrary point of interest at whih the potential is desired. Let Y ' be the image of Y in the plane fae reated by the dieletri boundary and rand r' be the distanes of point P from points Y and Y ' respetively. Assume that the line harge PL is plaed at Y and the field in the region Y > 0 is given by plaing at point Y ' a line harge -pl' suh that the potential is given by7 <P 1 =P L ln!. TI EO r Now suppose that the field inside the dieletri medium is due to a line harge PL" plaed at Y. Then 1 the potential = P "/TI ln satisfies Laplae's* L 0 r equation in the dieletri. The values of P L ' and P L " are determined by utilizing the boundary onditions at the interfae: 1. The potential is ontinuous at the interfae.. The normal omponent of the displaement vetor D is ontinuous at the interfae. Substitution of the potentials into the equations formed by the boundary onditions yields *The matter of satisfying Laplae's equation is sig~ifiant beause, for the mirostrip, it implies the assumpt~on of a TEM mode.

20 10 VACUUM z Figure 3.. ne Chargeb~ n Front of Filam7ntaDieletr1 Infin1te ry L1. Sla a semi I ' ~ p / y L ORIGINAL _jx CHARGE y. z VACUUM

21 11 p -pl=p" L L L ( 1) and 0El y = o r E y. ( ) or ( 3} Thus, (4) Solving Equations (1) and (4} simultaneously yields p L I 1 - = ( r) r ( 5) and ( 6) More generally, for a two dieletri system 1 and : PL I = ( - ) 1 + PL ( 7) and PL = II ( ) PL. ( 8)

22 1 3. Appliation of the Image Tehnique to the Mirostrip for Determining the Potential and Eletri Field a. Determination of the Charge Configuration The mirostrip problem is attaked by onsidering an axial element of the surfae of the upper ondutor as a filament of harge (see Figure 5). A suffiient number of image line harges of this filamentary harge satisfy- ing all of the boundary onditions, inluding those at the air-dieletri interfae and at the ground plane, is determined. From this known harge onfiguration the potential and eletri field at arbitrary points, both in and above the dieletri, is found. Finally, the eletri field resulting from the total strip is determined by integration about the periphery of the upper ondutor, whih effetively sums the ontributions of all of the harge filaments making up its surfae. Assumptions are as follows: 1. To simplify the alulations, the dieletri onstant of air is onsidered to be equal to Eo-. The extent of the dieletri in both the positive and negative X diretions is onsidered infinite. 3. The ground plane is of zero thikness and infinite in extent in X. 4. Attenuation is exluded from this analysis.

23 13 UPPER CONDUCTOR AIR Figure 5. z X AIR GROUND PLANE A Profile View of the Mirostrip Showing Point P Whih Represents an Axial Filamentary Charge on the Surfae of the Upper Condutor Figure 6. The First Stage of the Image Solution Showing the Fititious Dieletri and the Image of PL in the Air-Dieletri Boundary

24 14 Referring to Figure 5, an arbitrary point P on the upper ondutor surfae, loated in terms of X, y and z oordinates, is hosen for onsideration. Assume that the point P represents a line harge PL parallel to both the ground plane and the air-dieletri interfae. For all first approximations in the following work, the harge distribution about the periphery of the strip is onsidered onstant. Approahing the mirostrip problem as a ombination of the situations enountered in Setions III-A-1 and III-A-, the image harge onfiguration is thus determined. Referring to Figure 6, the ground plane is onsidered Y = 0, with the real dieletri extending a distane Y = a. The addition of a fititious dieletri extending a distane Y = a in the negative Y diretion symrneterizes the problem without affeting its final solution. The line harge PL lies a distane b above the dieletri or (a + b) above the ground plane. Applying the image tehnique for above the dieletri results in a first image -pl at a distane (a- b) above the ground plane. This image line harge results from the onsideration of the line harge PL in front of the dieletri at Y = a, as in Setion III-A-. On the other hand, a onsideration of the image of PL in the ground plane, as in Setion III-A-1, results in a seond image line harge -PLat a distane (a+ b) below the ground plane,

25 15 as shown in Figure 7. Carrying the treatment yet one step further alls for the refletion of the line harge -PL (itself an image) in the dieletri boundary at Y = +a to form an image line harge PL' at a distane of (3a +b) above the ground plane. This proess of refletion and re-refletion would seem to asade to inlude an infinite number of line harges both above and below the ground plane. However, the harge PL' at Y = (3a +b) above the ground plane ompletes the solution, sine the harge onfiguration shown in Figure 8 satisfies all boundary onditions of the problem. Removing the dieletri and the ground plane leaves only the line harges whose potential is given by lassial eletrostati theory7 as P L x + (Y_ +_a + b) =~ln[.. 0 X + (Y - a - b) P ' x L ln [ + (Y - 3a - b) 4rr 0 x + (Y- a+ b) (9) for Y > a. For the potential inside the dieletri region above the ground plane (0 ~ Y < a), the harge onfiguration is as shown in Figure 9. Again, the dieletri and ground plane may be removed, leaving only the two line harges PL" and -PL",

26 16 AIR T. Y = a+ b p L y - a AIR -p L y = -a- b Y = -a IMAGE OF ORIGINAL CHARGE IN GROUND PLANE Figure 7. Illustration of the Addition of the Image in the Ground Plane of the Original Line Charge PL

27 ItS M Figure 8. The Final Charge Configuration with the Third Image Charge Plaed at Y = 3a + b

28 18 AIR AIR! -p " L Figure 9. The Charg Inside e Conf" th~gu7ation D1eletri for the Potential

29 19 whose potential is given by eletrostati theory7 as PL" x + + ( Y a + b) = ln r ~-..:...;;;. ~-~~ 0 4 '1T 0 x + (Y - a- b) J ' < y < a (10) Thus, the potential for all points above the ground plane has been determined for an arbitrary axial filament of harge on the surfae of the rnirostrip ondutor. Hereafter, subsripts 1 and will designate regions above and inside the dieletri respetively. b. Determination of the Values of pl' and pl" The values of pl' and pl" are determined by utilizing the boundary onditions at the air-dieletri interfae. These are given by the expressions: Cl<Pl ra <P ay = ay Y=a Y=a ( 11) and <Pl = Y=a Y=a (1) The partial derivatives of 1 and with respet to Y are given by

30 0 Y + a + b [ X + (Y + a + b) Y - a - b [ Y - 3a - b Y - a + b ] x + (Y- 3a- b) x + (Y - a+ b) ( 13) and = Y + a + b Y - a - b [ x + ( Y + a + b) (14) Substituting Equations (13) and (14) into Equation (ll) and Equations (9) and (10) into Equation (1) yields the following equations: p + p I = E: p II L L r L (15) and (16) Solving Equations ( 15) and ( 16) simultaneously gives the following expressions for PL I and PL II 1 - E: r) PL I = -(1 + E: r PL (17) and (1 ) (18} + E:r II PL = PL.

31 1 The potentials an now be written as pl = PL x + (Y ln + a + b) 4n 0 [ X + (Y - a - b) 1 - PL x (Y - 3a - b) + (1 r) ln + + r 4n 0 x + (Y - a + b) ] ' y > a (19) and 4> = PL x + (Y ln + a + b) 1 [ ] 0 < + r 471' 0 X + (Y - a - b) ' - y < a. (0). The Eletri Field of the Filamentary Charge in the Region Above the Ground Plane The eletri field above the ground plane is determined by utilizing the expression E = -'V<jl, ( 1) where the del operator is defined in artesian oordinates in two dimensions by the expression () First it is onvenient to make a hange of variables whih will later allow an integration about the ondutor surfae to determine the total eletri field. Letting

32 X= (X- X') and b = Y', the expressions for the potentials beome: <1>1 = {X- X') + (Y + Y' + a) {1n [ J {X- X') + (Y- Y' - a) (X - X I) (Y - Y' r) ln[ + - 3a) ] Er (X - X I) + (Y + Y' - a) } <r + ( 3) and = (1 PL ln [ (X - X I) + (Y + Y' + a) + Er 4'TT 0 J. (X - X I) + (Y - Y' - a) The partial derivatives of 1 and with respet to X and Y are given by the expressions: ( 4) a 1 PL ax = 7T 0 X I ) a) { [ X - X' (X - + (Y + Y' + X - X' (X - X I) + (Y - Y' - a) J 1 - r) [ X - X' + (1 + X I) 3a) Er (X - + (Y - Y' - X - X' ]} (X- X') + (Y + Y' - a) ( 5)

33 3 a<pl PL y + y I { [ + a w- = 7fe:o X') a) (X - + (Y + Y' + y - Y' - a (X - X I) + (Y - yl - a) J 1 - e: + r) (1 + [ y - Y' - 3a e: r (X - X I) + (Y - Y' - 3a) y + Y' - a (X - X I) + {Y + Y' - a) ]} ' ( 6) a<p PL [ X - X' r (X - + (Y + Y' ax = 1 + e: 7fe:o X I) + a) X - X' J (7) (X - X I) + (Y - Y' - a) ' and a<p PL [ y + Y' + a w- = 1 + e:r 7fe:o (X - X I) + (Y + Y' + a) Y - Y' - a J (X- X') + (Y- Y' - a) ( 8) The eletri fields in regions 1 and (above and within the dieletri respetively) due to a filamentary line harge are given by substituting Equations {5) through (8) into Equations (1) through (4), and beome

34 4 {[ X - X' (X- X') + (Y + Y' + a) X - X' J (X - X') + (Y - Y' - a) X - X' (X - X') + (Y + Y' - a) ]} + a y {[ y + Y' + a (X - X') + (Y + Y' + a) y - Y' - a (X - X I) + (Y - Y' - a) J 1 - e: y r) - Y' - 3a e: r X') 3a) + <r + [ (X - + (Y - Y' - Y + Y' - a ]} (X- X') + (Y + Y' - a) (9) and

35 5 a [ x - x X (X- X') + (Y + Y' + a) X - X' (X- X') + (Y- Y' - a) ] y + a + Y' [ + a y (X - X I) + (Y + Y' + a) y - Y' - a (X - X I) + (Y - Y' - a) J, (30) where the onstants A and B are defined by A PL and 1 PL = B TT O = 1 ( 31) + r TT 0 d. Determination of the Eletri Field of the Upper Condutor in the Region Above the Ground Plane Referring to Figure 10, the primed oordinate loates the filamentary harge; the unprimed oordinates refer to the arbitrary point R at whih the eletri field is being determined. The width and thikness of the strip are defined as ~ and t respetively. The total eletri field ontributed by all of the filamentary harges making up the surfae of the ondutor is determined by summing all of their individual vetorial ontributions. This operation is ahieved by integrating the filamentary field expressions in the primed oordinate

36 6. y'= 0 Figure 10. Illustration of the Primed Coordinates and the Dimensions and Boundaries of the Upper Condutor (Points R and P Lie in the Plane of the Page)

37 7 system about the ondutor boundary. A general expression of this operation is given by Y'=t + I de(x' = ~) Y'=O + x r/ X'=- / de(y' = t) Y'=t + de(x' = } I Y'=O -9., (3) or ET = EA + EB + E + ED (33} where the subsripted fields are the individual surfae ontributions defined by the integrals in Equation (3}. Use of the inremental harge onept gives rise to the relabeling of the eletri field in expressions (9) and (30} from E 1 and E to de1 and de respetively. Substitution of de1 and de into Equation (3} gives the ontributions of eah of the four ondutor sides to the total eletri field. These are as follows:

38 8 In region 1, E1 A x r/ (( X - X' A = a X X'=-.R-/ X I) a) (X - + (Y - X - X' J (X - X') + (Y + a) [ X - X' (X- X') + (Y - a) X - x J} dx' (X - X I) + (Y - 3a) + a y X'=.R-/ { [ y - a X I) f (X - + (Y - a) X'=-.R-/ Y + a ] (X- X') + (Y +a) 1 - y - a ( r) [ r (X- X') + (Y- a) Y - 3a ] } dx', (X- X') + (Y- 3a) (34)

39 E1 B p:- = ax r X Y'=t Y'=O {[ R. - R. ) a) (X - + (Y - Y' (X - R. X -!) + (Y + Y' + a) J (1 + e:r e: ) [ r (X - R. X - ~) + (Y + Y' - a) (X - R. X-!) + (Y- y' - 3a) ]} dy' Y't Y - Y' - a { [ (X - i> + ( Y - Y' - a) Y'=O y + Y' + a (X - ~) + (Y + Y' + a) J e: y + Y' - a r) [ (1 + e: r ~) (Y + Y' a) (X y - Y' - 3a ]} dy' i) + (Y _ y - 3a) (X - ( 35)

40 30 a X x r/ {[ X'=-R./ x - x (X- X') + (Y- t x - x ~---~ ~] (X- X') + (Y + t + a) 1 - r X - X' + ( 1 + r) [ (X - X' ) + ( Y + t - a) X - x ]} dx' (X - X I) + (Y - t 3a) + a y X'=R./ I {[ X'=-R./ y - t - a (X- X') + (Y - t a) y + t + a (X - X I) + (Y + t + a) J 1 - y + t - a r) + (1 + [ r (X - X I) + (Y + t - a) y - t - 3a dx' ]} (X - X I) + (Y - t - 3a) I ( 36) and

41 31 a X Y't Y'=O X +! {[ (X+ ~) + (Y- Y' - a) (X + X +! + Y' + a) J X + 5I, ]} dy' (X+ ~) + (Y- Y' - 3a) Y'=t y - + a f { [ - Y' a y 51,) (X + + (Y - Y' - a) Y'=O y + Y' + a J (X +!) + (Y + Y' + a) 1 - E: y + Y' + r) [ - a (1 + E: r (X +!) + (Y + Y' - a) Y - Y' - 3a ]} dy' (X+!) + (Y- Y' - 3a) ( 3 7) Similarly, the expressions for region are

42 3 X'=R./ X - X' f [ _(_X X_'..:.;);., +..::.:...(_Y a_)""'l'r' X'=-R./ ~-X' (X- X') + (Y +a) ] dx' + a y x r/ [ Y - a X'=-R./ (X- X') + (Y- a) Y + a J dx', (X- X') + (Y + a) ( 38) E B = B a X Y't Y'=O [ X R. - (X -!) + (Y - Y' - a) R ,.r-x_- ::= ;;- J dy ' (X - }> + (Y + Y' +a) + a y Y - Y' - a [ (X _!) + ( y _ y, _ a) Y + y + a ] dy', (X-!) + (Y + Y' + a) (39)

43 33 E """"13 = a X x r/ X'=-./ [ X - X' X I) a) (X - + (Y - t X - X' (X - X I) + (Y + t + a) ] dx' + a y x r/ y - t - a X'=-./ [ (X - X I) + (Y - t - a) y + t + a (X - X I) + (Y + t + a) J dx' I (40) and a X Y' =It [ ---;;;---.o..--x_+_;:::.. ""5" (X+ }> + (Y- Y' - a) Y'=O Q, X r;---.oor-- ;::::? (X + ~) + (Y + Y' + a) J d y I y I =ftc [ ---;;.._..,~:.....:::;_ y - Y' - a, + a y Y'=O (X+ ~) + (Y- Y' - a) Y + Y' + a ] dy' (X+!) + (Y + Y' +a) ( 41)

44 34 Performing the integrations gives rise to the eletri field expressions due to the respetive sides of the upper ondutor. The ontribution of the bottom surfae is: El A Ao ax = { ln [(X -!.) [(X +!.) + (Y + a) ] [ (X +!) + (Y + a) ] [ (X!) + (Y - + (Y - a)] a) ] [ (X-!.) ln [ (X+!_) (Y-a) ] (Y-a) ] } ~ ~ ~ X- X+ X+ + ay {Artan Y + a - Artan y + a + Artan y _ a x-- 1-E - Artan Y _ a + ~ ~ Er 9- R. ( r) X - (Artan y _ Ja 9- R. X + X + - Artan Y _ Ja + Artan y _ a ll, X - - Artan Y _ a ) }, (4) where the onstants A 0 and B 0 whih appear in this and subsequent expressions are defined by and B 0 = Er The eletri field ontributions of the remaining surfaes are given by:

45 35 El y + t + B a {Artan y + a Ao = a - Artan X R. X R. - X - + Artan y - t - a R. + Artan y - a R. X - X - - Artan Y ----~~-- + t - a +Artan Y- a)} X _R. X R. - R. R, a [(X--) + (Y+t+a) ][(x- > + (Y-a) ] + ~ {ln----~--~----~~~--~r-~~ ~-- [(X-~) + (Y+a) ][(X-) + (Y-t-a) ] 1-Er [(X-~) + (Y-t-3a) J[(X-~) + (Y-a) ] + (1+ ) (ln----~r-~~------~=-----~r-~~ ~- r [ (X- ) + (y-3a) ][ (X- ) + (Y-t -a) ] ) } I ( 4 3)

46 36 [(X-!_) + {1n [(X+~) + (Y+t+a> J[(X+~) (Y+t+a) J[(X-~) + (Y-t-a) J + (Y-t-a) J ~ (Y-t-3a) ] [ (X+- ) + (Y+t -a) ] ) } (Y-t -3a) J[(X-!) + (Y+t -a) J ~ ~ x- X+ + ay {Artan Y + a + t - Artan y + a + t x-! x+! - Artan ~Y~---a---~t- + Artan y _ a _ t ~ ~ 1- x- X+ + ( 1 r) (Artan Y - Artan y _ Ja _ t + r - 3a - t ~ X + ~ X - -Artan t +Artan y Y - a + _a+ t )} ' (44) and

47 37 El D --- = ax {Artan Ao Y + t + a ~-- - Artan Y + a 1 1 X+~ X+' - Artan Y - a - t X + t + Artan Y - a t X + Y - t - 3a X Artan Y - 3a 1 X+ -Artan Y + t - a +Artan Y- a)} X + t X + t t t 1-e:r [ (X+) + (Y-t-3a) ] [ (x+) + (Y-a) ] + ( l+e: ) ( ln 1 t ) } r [(x+) + (Y-3a) ][(x+) + (Y-t-a) ] ( 4 5) Similarly, in the dieletri,

48 38 E a [ex-!> + (Y+a) J[(X+~) A X [ln Bo = [(X+~) + + (Y-a) ] (Y+a) J[(x-;> + (Y-a) J J X - X + R- + a [Artan - Artan y Y + a Y + a R- t ~ x- X+- - Artan Y _ a + Artan Y _ : ], ( 46) E y + B t + a y [Artan + - Artan a Bo = ax R- t X - X - - Artan y - t - a + Artan y - a R- R- X - X - J a [ (X-!) + (Y+t +a) J[(x-!> + (Y-a) ] +-r [ln J I [(X-~) + (Y+a) J[(X-~) + (Y-t -a) ] ( 4 7) E [(X-~) (Y+t+a) J[(X+~) a + + (Y-t -a) ] ~ [ln = Bo R- [(X+~) + (Y+t+a) ][ (X-) + (Y-t -a) ] J x-r- x+r- + a [Artan Y + t + a - Artan y + t + a y t t X - X + - Artan Y _ t _ a + Artan y _ t _ a J ( 48) 1

49 39 and, Y + t + a [Artan - Artan Y + a X + ~ X +.II. - Artan a +.,_Y Y - t - a X+! + Artan Y - a.ii. X+ (49) Thus, the eletri field has been determined for all points in spae about the mirostrip ondutor. In the dieletri it is the sum of expressions (46) through (49) and above the dieletri the eletri field is determined by summing expressions (4) through {45). e. Determination of the Potential at a Point Above the Ground Plane Due to the Conduting Strip Although the potential is not neessary for the primary objetive of this paper, the impedane, it is a valuable quantity in many analyses and omes as a byprodut of the present work. The potential at a point due to the total onduting strip may be determined by summing the potential ontributions of all of the filamentary line harges making up the

50 40 periphery of the upper ondutor, in a manner similar to that presented in Setion III-A-3-d of this paper. Equation (50) is a statement of this operation: = x r/ X'=R./ 4> (Y I = f 0) + x =r.x. =- R./ R. Y'=Jt = -) + ~f(x' Y'=O =!.) (50) When evaluating expression (50) in region 1, ~f beomes ~l as given in expression (3), and in region, jlf beomes 4> as given in expression (4). After performing the integrations, the potentials are found to be: [ <x-l> + ln [ <x-l) + Jl, (Y-a) 1 [(X--) + (Y+a) 1 [ex-!> + (Y-t -a) 1 (Y+t +a) 1 + Artan y + t + a Jl, X - - Artan y + a R. X - + Artan y - t - 1 X - a - Artan y - a Jl, X - 1

51 41 + (X+~) [ ~ (Y+t +a) ] (y-t -a) ] Y + t + a + Artan - Artan Y + a R. X+ X+ + Artan y - t - a y - a - Artan R. X + X + J R. R. X + X - + (Y+a)[Artan - Artan y + a y + a R. R. X - X + + (Y-a)[Artan y _a- Artan y _a R. R. X+ x- + (Y+t+a)[Artan Y + t +a- Artan y + t +a

52 4 + Artan y X y 3a - t - 3a - - Artan - X - y y + t - a - a + Artan Q; - Artan X - X - J X + X - + (Y-3a) [Artan Y - 3a - Artan y _ 3a 1 + ~ ln [(X-~) + (Y-3a) ][(X+) + (Y-3a) ]] x- X+ + (Y-a)[Artan Y - a -Artan Y - a 1 R, + 1n [ (X+~) + (Y-a) ] [ (X- ) + (Y-a) ] ]

53 43 i i X+ x- + (Y-t-3a)[Artan Y _ t _ Ja- Artan Y _ t _ 3 a i X - + (Y-t -3a) ]] + (Y+t -a)[artan ~~+~t----- Y - a X + R.. 1 i - Artan Y + t _ a - ln [ (X- ) (51) the potential in the region above the dieletri, and Y + t + a Y + a + Artan - Artan i X + ~ X + Y - t - a + Artan - Artan Y - a ] R.. X + i X+ [(x-!> + (Y-a) J[(x- R..> + (Y-t -a> J i 1 + (X-)[ ln-[-(-x--~}-)~--+ (_Y_+_a_)~-J-[-(X--~~-)~-+---(Y_+_t +_a_)~r-j

54 44 y + Artan + t + a X Jl.. - Artan y + a Jl.. X - - y - t - a + Artan y - Artan - a Jl.. X - X - Jl.. J X + ~ X - Jl.. + (Y+t+a)[Artan Y + t +a- Artan Y + t +a X+ X-- + (Y+t+a) ]] + (Y+a)[Artan Y +a- Artan Y +! Jl.. Jl.. 1 J/.. J/.. - ~ 1n [(X+ } + (Y+a) ][(X-~} + (Y+a} ]] x- X+ + (Y-a}[Artan -Artan y _ Y - a a Jl.. Jl.. 1 J/.. J/.. - ~ 1n [ (X-~} + (Y-a} ] [ (X+") + (Y-a) ] ] X Jl.. - X + ~ ~ + (Y-t -a)[artan Y -Artan y t - t - a - - a (5)

55 45 the potential in the dieletri. With the expressions developed thus far, suffiient information is available for preparation of equipotential skethes and eletri field plots for a mirostripline having any ombination of dieletri thikness, ondutor thikness and ondutor width. B. Expressions for Capaitane and Impedane 1. A First Approximation to the Capaitane Had it been possible at the onset of the problem to state the variation of pl with X and Y on the strip, the expressions (4) through (49) would give exat values for the eletri field. Sine this variation is yet unknown, it is assumed to be onstant for all first approximations. In the first approximation to the apaitane and line impedane, the apaitane per unit length is omputed from the expression Q = - (53) I where Q is the harge per unit length on the strip and is the potential between the strip and the ground plane. The total harge on the strip is determined by summing all of the elements of harge about its surfae perimeter and is expressed by

56 46 p(y = a)dx + s -!G/ -!G/ Q = J Ps ds = T tf p(y = a+ t )dx + r t r t Q, p(x = -)dy + p(x = ) dy. (54) 0 0 Q, Sine the harge distribution is assumed onstant, the expression for harge per unit length beomes simply (55) As a result of the assumption that the onduting strip is a perfet ondutor, the potential at all points along the strip is equal. For simpliity, the potential will be omputed between the onduting strip and the ground plane at the point X = 0. The expression for the eletri field in the dieletri at X = 0 will beome useful in this derivation. It is simply the summation of the ay omponents of expressions (46) through (49). The expression for potential between two points is b -+- = J E dq. "'ab a (56) Equation (56) beomes, after substitution of Equations (46) through (49) at X = 0,

57 Y=a ne:o< 1 +e:r) j p = [Artan 9-/ + Artan Y 9-/ PL y - a - t - a Y=O - Artan 9-/ 9-/ y + a - Artan Y + t + a 47 Y+t +a Y-t -a + 1n v1 +( ) + 1n 11 + ( ) t/ 9-/ - 1n J1 + (Y-a) - 1n "1 + (Y+a) Ja m m y a dy ( 56a) y where d is a dy. y After integration, the expression beomes [1 + (t/t) ] ] [1 + (4a/ ) ][1 + (a+t /(9-/)) ] + [ ~(1 + Arot :- 1n V1 + ( ~ ) :a ( + 1-1n V Arot 4a),Q, a+t + ( t/ )./ a+t (1n V 1 + ( 7 ) - 1- Arot a+t / ) 4a - Artan y- + Artan a+t t /- Artan~]. (57)

58 48 redues to For the strip of zero thikness, expression (57) 1<1>1 = (58) Substitution of the results of Equations (55) and (57) into Equation (53) gives an expression for the mirostrip apaitane as a funtion of measurable physial parameters. For the zero thikness ondutor, the final expression for the first approximation to the apaitane beomes (59) Figures 11, 1, 13, and 14 are plots of the mirostrip apaitane for various relative dieletri onstants and ondutor width to dieletri thikness ratios. Examination of expression (57) indiates that while the value of the apaitane is affeted by the inlusion of t, the ondutor thikness, its effet is not appreiable for small thiknesses.. Higher Order Approximations to the Capaitane Expression (59) is alled a first approximation to the mirostrip apaitane beause it is based upon the assumption that the variation of pl aross the ondutor

59 ZERO THICKNESS CONDUCTOR APPROXIMATION (/) Q <( ei 500 u.. 0 u... Q. 400 I u z ~ u <( Q Sy: -::: l Figure /a - CONDUCTOR WIDTH-TO-DIELECTRIC-fHICi~ESS-RATIO Mirostrip Capaitane Versus Condutor Width-to-Dieletri Thikness Ratio (For 1/a Values 1-15 and r Values 1-10) ~ 1.0

60 6 ZERO THICKNESS CONDUCTOR APPROXIMATION 5 ~ 4 < 0::: < u. 0 ~ 3 z I L1J u ~... -u < 0.. < ull- 'l"~~~ ~ ". ~ 10 Figure 1. tiirostrip Capaitane Versus Condutor Width-to-Dieletri Thikness Ratio (For ~/a Values 1-15 and Er Values 10-90) U1 0

61 5 r I I I I I I I I I -G 4 ZERO THICKNESS CONDUCTOR APPROXIMATION (/) Q <( 0:::: <( LL. 0 z <( z I w u z <( u <( 0... <( u 3 1 Figure t/a - CONDUCTOR WIDTH-TO-DIELECTRIC THICKNESS RATIO Mirostrip Capaitane Versus Condutor Width-to-Dieletri Thikness Ratio (For t/a Values and Values 1-10) r lj1 1-'

62 40 35 ZERO THICKNESS CONDUCTOR APPROXIMATION 30 (/) Q <C a:: <C u. 0 z <C z I 5 0 UJ 15 u z <C u 10 <( a. <( u 5 Figure /a - CONDUCTOR WIDTH-TO-DIELECTRIC THICKNESS RATIO Mirostrip Capaitane Versus Condutor Width-to-Dieletri Thikness Ratio (For 1/a Values and Er Values 10-90) U1 1\J

63 53 ;s on t t Y h"t 14 "d s an amas ~ a ons~ ered th~s problem in his determination and onluded that for the type of variation whih exists in the mirostrip geometry, the assumption of a onstant harge variation has little effet on the impedane (and onsequently the apaitane) determination. Beause the results of the impedane determination of this paper are aeptable, higher order approximations are onsidered unneessary for inlusion in the final apaitane and impedane presentations. If desired, however, higher order apaitane approximations are possible through an iterative proess arried out in the following manner: From the determination of the eletri field based upon a onstant pl variation, the flux density is found. Values of pl(x,y) determined from the flux density evaluated at the ondutor surfae are used to obtain a new set of values for the eletri field through a reappliation of the image tehnique presented in Setion III-A. When iterations on this proess onverge, the final potential differene between the ground plane and the ondutor and the total harge on the ondutor are used to determine the value of the mirostrip apaitane. 3. The Impedane Problem In ideal transmission line theory, the ratio of the potential differene between two ondutors to the

64 54 total urrent flowing on the ondutors for a wave propagating in either the positive or negative diretion is the harateristi impedane z of the line z = where Z = (~/:) 1 1 :z = =- ( 6 0) ( 61) is the intrinsi impedane of the medium surrounding the ondutors, and C is the eletrostati apaitane. Thus, for a homogeneous dieletri extending to infinity, the harateristi impedane differs from the wave impedane by a apaitane fator whih is a funtion of the geometry only. In the theoretial mirostrip problem the impedane determination is ompliated by the multi-dieletri harater of the medium in whih the wave is traveling. In addition, the pratial mirostrip dieletri is finite in width, whih additionally ompliates the determination. Thus, the intrinsi impedane in expression (60) annot In short, the problem is what value should be used for the dieletri onstant in expression (61) when the wave in a mirostrip travels in a multi-layered medium. Wheeler10 enountered this problem in his work and proeeded to develop a "field form fator" and a related effetive dieletri onstant whih he used to adjust

65 55 his expressions to meet the measured values. Sekelmann8 also beame aware of this problem and proeeded to take laboratory measurements in an attempt to verify Wheeler's work. An examination of impedane alulations determined through the use of the expression (JJ )1/ 0 r I ( 6 ) where (JJ ) 1 / is the speed of light in the dieletri material, indiates that its use by many investigators, inluding Yamashita, is invalid for the mirostrip onfiguration. This disagreement is quite evident in Figure 15. Cursory analysis of this and similar plots would perhaps indiate that the analysis presented here is merely off by some onstant... f h d d b K ' zg Exam1nat1on o t e ata presente y a1ser suggests an approah in terms of the propagation onstant. Determination of the harateristi impedane of the mirostrip through the use of the expression, ( 6 3) where r is the propagation onstant, gives rise to data suh as that found in Figures 16 and 17. Examination of this data shows very lose agreement with Wheeler and good agreement with Kaiser's experimental data. Kaiser gives no data onerning the ondutor

66 " 140 t \ a \ =.065 Er =.6 I \ \. 10 (/) ~ 100 t \ ~ UJ u z <t Q UJ 80 tl.. :::E - 60 u ~ / (/) MEASURED 40 VALUES 0::: UJ 1- u <t 0::: <( ::r: u 0 RAW.,.. CALCULATED VALUES (EQUATION (6)) Figure ~/a - CONDUCTOR WIDTH-TO-DIELECTRIC THICKNESS RATIO A Comparison of Raw Theoretial and Experimental Impedane Values (JI 0'1

67 10 -en ~..., MEASURED PROPAGATION CONSTANT MEASURE~ IMPEDANCE WHEELER S MODIFIED SCHWARTZ-CHRISTOFFEL IMPEDANCE BY IMAGE IMPEDANCE TECHNIQUE 1- I l!)......i u. 0 UJ u ~ 80 Q UJ a.. ~ - 60 u -len -0::: ~ 40 u ~ 0::: ~ I u " DIELECTRIC THICKNESS (a) =.065 e:r =.6 1 Figure CONDUCTOR WIDTH-TO-DIELECTRIC THICKNESS ~ - CONDUCTOR WIDTH (INCHES) Theoretial and Experimental Impedane Plots 7 RATIO (a = 1/16") Q UJ 0 a.. UJ en u z UJ u 0::: UJ a.. I 1- z ~ 1- en z 0 u z ~ l!) ~ a.. 0 0::: a.. U

68 10 I \ MEASURED PROPAGATION CONSTANT MEASURED IMPEDANCE \ - WHEELER'S MODIFIED SCHWARTZ-CHRISTOFFEL -~ soi ---- IMPEDANCE BY IMAGE IMPEDANCE fo 1- -, r TECHNIQUE : en _J : hn 0..., UJ.. u en z 60 <( LL Q 0 UJ z ~... UJ 40 u u... 0::: 1- UJ en '1"'\.'1'~1 ~,..-rr""\ti""'.,..iiti"'ij'~ir""l"'l"' ('!"1\ ::: 0 UJ t u Er =.6 z <( <( 10 0::: 1- <( en : z u 0 u z i/a - CONDUCTOR WIDTH-TO-DIELECTRIC THICKNESS RATIO ~ <(.. 0 0::: Figure 17. Theoretial and Experimental Impedane Plots (a = 1/3").. i - CONDUCTOR WIDTH (INCHES (!)... LL Q LlJ LlJ 1- <( (!) V1 (X)

69 59 thikness or the dieletri width. In the alulations the ondutor thikness was onsidered zero. Analysis of the expression indiates that an impedane differene of no more than one or two ohms ould be expeted with typial ondutor thiknesses.

70 60 IV. CONCLUSIONS, DISCUSSION AND SUGGESTIONS FOR FURTHER WORK A. Conlusions and Disussion The results presented in this paper are in reasonable agreement with the literature and ertainly suggest the validity of the image tehnique. The assumption of a TEM mode through the appliation of the Laplae equation in the assumptions neessary for the appliation of the image tehnique is regrettable from the standpoint of rigor but is unfortunately neessary. This assumption has preipitated omment from various 7 investigators, among them Dukes, who reognized that, although the mode is not TEM, the assumption of a TEM mode produes valid results. Certainly, its suess in this paper does not weaken the ase for use in further work. The suess of the impedane determination lends redene to the validity of the apaitane determination (although no data an be found in the literature for substantiation) as well as the potential and eletri field determinations. The effet of geometry and size as well as the dieletri onstant upon the propagation onstant annot presently be stated with assurane, sine no theoretial

71 61 work an be found in this area. It is felt, however, that sine the propagation onstant is dependent upon these parameters, some margin for error exists when using experimental values. Of ourse, a thorough theoretial determination requires a look also at the propagation onstant from a theoretial standpoint. B. Suggestions for Further Work Certainly regrettable is the fat that a dearth of experimental work is available for examination. One of the shortomings of the present bit of work in the literature is that some data has been taken by one investigator with one method and a seond bit of data has been taken by another investigator by another method, neither one of whih an be ompared for auray or orretness. It is suggested that a omprehensive experimental study be undertaken to determine the apaitane, impedane, and propagation onstants of mirostriplines for a wide range of relative dieletri onstants and ~/a ratios. (To larify the issue, UMR presently has neither the tehnology nor the support to fabriate mirostriplines with suffiient quality ontrol to do a worthwhile study.) Further theoretial work should be done to determine the propagation onstant for waves traveling in the multilayered dieletri onfiguration peuliar to the mirostrip geometry.

72 6 If it is felt neessary, the iterative proess for upgrading the approximations presented in this paper ould be attempted by future investigators. Certainly, the work ould not be done in losed form but rather would require numerial tehniques. Moving beyond the harateristi impedane problem for the simple mirostrip onfiguration, there exist a number of mirostrip onfigurations whih need both theoretial and experimental work, inluding the oupling problem for parallel strips and the determination of the effets of stubs and tapers in mirostrip iruits.

73 63 VITA Joseph Louis Van Meter was born on Otober 8, 1945, in Maplewood, Missouri. He reeived his primary and seondary eduation in the shools of the Maplewood Rihmond Heights shool distrit in St. Louis County, Missouri. His undergraduate work was done at the Univer sity of Missouri - Rolla, in Rolla, Missouri. During this period, he spent alternate semesters at MDonnell Douglas Corporation, St. Louis, Missouri, where he was employed as a member of the Engineering Co-op Program. He reeived a Bahelor of Siene degree in Eletrial Engineering from the University of Missouri - Rolla, in Rolla, Missouri, in June He has been enrolled in the Graduate Shool of the University of Missouri - Rolla sine July He is a member of Tau Beta Pi, Eta Kappa Nu, Phi Kappa Phi, and the IEEE.

74 64 BIBLIOGRAPHY 1. Alford, Andrew, and Makay Radio and Telegraph Corp., U. s. Patent,159,648, May 3, 1949 (first filed in 1937}.. Frieg, D. D., and Engelmann, H. F., "Mirostrip-A New Transmission Tehnique for the Kilomegayle Range," Pro. of the IRE, 40 (195}, Assadourian, F., and Rimai, E., "Simplified Theory of Mirostrip Transmission Systems," ibid., p Kostriza, J. A., "Miros trip Components," ibid., p Arditi, M., "Charateristis and Appliation of Mirostrip for Mirowave Wiring," IRE Trans. on Mirowave Theory and Tehniques, MTT-3 (Marh-,- 1955) 1 P Blak, K. G., and Higgins, T. J., "Rigorous Determination of the Parameters of Mirostrip Transmission Lines," ibid., p Dukes, J. M.., "An Investigation Into Some Fundamental Properties of Strip Transmission Lines With the Aid of an Eletrolyti Tank," Pro. IEE (London}, Vol. 103, Pt. B (May, 1956), pp Wu, T. T., "Theory of the Mirostrip," Journal of Applied Physis, 8 (1956}, pp Kaupp, H. R., "Charateristis of Mirostrip Transmission Lines," IEEE Trans. on Eletroni Computers, EC-16 (1964), pp~5-l Wheeler, H. A., "Transmission Line Properties of Parallel Wide Strips by a Conformal Mapping," IEEE Trans. on Mirowave Theory and Tehniques, MTT-1 ( }, pp Wheeler, H. A., "Transmission Line Properties of Parallel Strips Separated by a Dieletri Sheet," IEEE Trans. on Mirowave Theory and Tehniques, MTT-13 (1965); pp

75 Stinehelfer, Harold E., Sr., "An Aurate Calulation of Uniform Mirostrip Transmission Lines," IEEE Trans. on Mirowave Theory and Tehniques, MTT-16 (1968),-p Yamashita, E., "Variational Method for the Analysis of Mirostrip Lines," IEEE Trans. on Mirowave Theory and Tehniques,:MTT-16 (1968}, p. 51. Yamashita, E., "Variational Method for the Analysis of Mirostrip-Like Transmission Lines," IEEE Trans. on Mirowave Theory and Tehniques, MTT-16 (1968), p Hallen, Eri, Eletromagneti Theory, New York: Wiley, 196, pp Ferrer, A., and Adams, A. T., "Charateristi Impedane of Mirostrip by the Method of Moments," IEEE Trans. on Mirowave Theory and Tehniques, MTT-18 (l970f; pp Harrington, R. F., Field Computation~ Moment Methods, New York: MaMillian, Chen, w. H., "Even and Odd Mode Impedane of Coupled Pairs of Mirostrip Lines," IEEE Trans. on Mirowave Theory and Tehniques, MTT -18 (1970f; p. 55. Krage, M. K., and Haddad, G. I., "Charateristis of Coupled Mirostrip Transmission Lines," IEEE Trans. ~ Mirowave Theory and Tehniques, MTT-18 (1970) I P 17. Maesel, M., "A Numerial Solution of the Charateristis of Coupled Mirostrips," Proeedings of the Eurolean Mirowave Conferene, London, England TI970), p Standley, R. D., and Braun, F. A., "Experimental Results on a Millimeter Wave Mirostrip Down Converter," IEEE Trans. on Mirowave Theory and Tehniques, ~-18 (1970f; p. 3. vane, A. B., and Dunn, V. E., "A Digitally Tuned Gunn Effet Osillator," ~ of the IEEE, 58 (1970), p Heht s., Kuepis, G. P., and Taub, J. J., "A I lt II Mirostrip Diplexer Using Bandstop F~. ers,. Proeedings of the 1969 Miroeletron~s Sympos1um, layton, Missouri; IEEE Publiation, p. a.5.