Scholars' Mine Masters Theses Student Theses and Dissertations 1970 A study of microstrip transmission line parameters utilizing image theory Joseph Louis Van Meter Follow this and additional works at: https://scholarsmine.mst.edu/masters_theses Part of the Electrical and Computer Engineering Commons Department: Recommended Citation Van Meter, Joseph Louis, "A study of microstrip transmission line parameters utilizing image theory" (1970). Masters Theses. 7119. https://scholarsmine.mst.edu/masters_theses/7119 This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. A STUDY OF MICROSTRIP TRANSMISSION LINE PARAMETERS UTILIZING IMAGE THEORY BY JOSEPH LOUIS VAN METER, 1945- A THESIS Presented to the Faculty of the Graduate School of the UNIVERSITY OF MISSOURI - ROLLA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE IN E~ECTRICAL ENGINEERING T2563 1971 c.2 pages 3 !J~ £ £~ (Advisor) ~d~4' (/ ~9424? ii ABSTRACT This paper is a theoretical investigation of the potential, electric field, capacitance, and characteristic impedance of the open strip transmission line or micro­ strip configuration based upon the classical Thomson Image Technique. It provides the basis for determination of the charge distribution on the strip and reports impedance values which compare favorably both with experimental values and theoretical work in the current literature. iii ACKNOWLEDGEMENT The author wishes to sincerely acknowledge the assistance and guidance given him by Dr. James Adair. His readiness to provide time for helpful discussion, suggestions and comments will always be remembered. iv TABLE OF CONTENTS Page ABSTRACT ii ACKNOWLEDGEMENT iii LIST OF ILLUSTRATIONS vi LIST OF SYMBOLS viii I. INTRODUCTION 1 II. HISTORICAL REVIEW 2 A. Literature Review 2 B. State of the Art 4 c. Object of Investigation 5 III. THEORY 6 A. Determination of the Electric Field and Potential by the Image Technique 6 1. The Line Charge in Front of a Ground Plane 6 2. The Potential for a Line Charge in Front of a Dielectric Slab 8 3. Application of the Image Tech­ nique to the Microstrip for Determination of the Potential and Electric Field 12 a. Determination of the Charge Configuration 12 b. Determination of the Values of p • and p " 19 L L c. The Electric Field of the Filamentary Charge in the Region Above the Ground Plane 21 v Page d. Determination of the Electric Field of the Upper Conductor in the Region Above the Ground Plane 25 e. Determination of the Potential at a Point Above the Ground Plane due to the Conducting Strip 39 B. Expressions for Capacitance and Impedance 45 1. A First Approximation to the Capacitance 45 2. Higher Order Approximations to the Capacitance 48 3. The Impedance Problem 53 IV. CONCLUSIONS, DISCUSSION AND SUGGESTIONS FOR FURTHER WORK 60 A. Conclusions and Discussion 60 B. Suggestions for Further Work 61 VITA 63 BIBLIOGRAPHY 64 vi LIST OF ILLUSTRATIONS Page 1 Filamentary Line Charge Above an Infinite Ground Plane 7 2 Profile View of the Ground Plane and Line Charge With the Image Line Charge in Place 7 3 Filamentary Line Charge in Front of a Semi-Infinite Dielectric Slab 10 4 Profile View of the Dielectric, Inter­ face, and Line Charge With the Image Line Charge in Place 10 5 A Profile View of the Microstrip Showing Point P Which Represents an Axial Fila­ mentary Charge on the Surface of the Upper Conductor 13 6 The First Stage of the Image Solution Showing the Fictitious Dielectric and the Image of pL in the Air-Dielectric Boundary 13 7 Illustration of the Addition of the Image in the Ground Plane of the Original Line Charge pL 16 8 The Final Charge Configuration With the Third Image Charge Placed at Y = 3a + b 17 9 The Charge Configuration for the Potential Inside the Dielectric 18 10 Illustration of the Primed Coordinates and the Dimensions and Boundaries of the Upper Conductor 26 11 Microstrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For ~/a values 1-15 and ~r values 1-10) 49 vii Figure Page 12 Microstrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For ~/a values 1-15 and £r values 10-90) 50 13 Microstrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For ~/a values 5-100 and £r values 1-10) 51 14 Microstrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For ~/a values 5-100 and £ values 10-90) r 52 15 A Comparison of Raw Theoretical and Experimental Impedance Values 56 16 Theoretical and Experimental Impedance Plots (a = 1/16") 57 17 Theoretical and Experimental Impedance Plots (a = 1/32") 58 viii LIST OF SYMBOLS A pL/2TI€0 a The Thickness of the Microstrip Dielectric a X A Unit Vector in the X Direction a y A Unit Vector in the Y Direction B pL/TI€0(l + €r) b The Height of the Filamentary Charge Above the Air-Dielectric Interface c Capacitance An Electric Field Vector A Total Electric Field Vector The Electric Field in Region 1 The Electric Field in Region 2 The Microstrip Electric Field Contributed by the Bottom Surface of the Upper Conducting Strip The Microstrip Electric Field Contributed by the Right Vertical Surface of the Upper Conducting Strip The Microstrip Electric Field Contributed by the Upper Surface of the Upper Conducting Strip The Microstrip Electric Field Contributed by the Left Vertical Surface of the Upper Conducting Strip The Width of the Upper Conductor of the Microstrip p An Arbitrary Point of Interest at Which a Potential or Electric Field Value is Being Determined Q The Charge Residing on the Upper Conductor of the Micros trip r The Distance From Point P to pL ix r' The Distance From Point P to the Image Line Charge The Thickness of the Upper Conductor of the Micros trip X The Width Coordinate of the Microstrip y The Height Coordinate on the Microstrip z The Length Coordinate on the Microstrip z Wave Impedance (Bar Distinguishes Between Impedance and Z Coordinate) z Characteristic Impedance of the Microstripline c (The Phrase "The Impedance" Unless Otherwise Stated Refers to the Characteristic Impedance) (The Bar Distinguishes from the Z Length Coordinate) r Propagation Constant An Arbitrary Dielectric Constant -9 The Permittivity of Free Space - l/36rr x 10 Farad/ Meter E: The Relative Permittivity of an Arbitrary Material r The Permittivity in Region 1 s 2 The Permittivity in Region 2 A Line of Filamentary Charge p I An Image Filamentary Charge Resulting From the Air­ L Dielectric Boundary p II An Image Filamentary Charge Resulting From the L Ground Plane Potential Function Potential Due to a Filament of Charge Potential Due to the Total Conducting Strip Potential Function in Region 1 Potential Function in Region 2 00 Infinity 1 I. INTRODUCTION Recent work in the field of semiconductor physics has brought to reality a suitable line of solid state devices which operate in the gigahertz range. To comple­ ment this ever growing group of active devices, the designer has looked to the stripline and microstrip configurations as a passive interconnection, thus scrap­ ping the old-fashioned waveguide plumbing and coaxial connections for more convenient structures. In addition, the microstrip holds some promise to the computer designer, whose objective is to minimize the transmission times of impulses traveling between interconnected devices. Fortunately, the study of microminiaturization techniques, which is necessary for the thin and thick film technologies, has also recently been accelerated. This makes it possible to exert a high degree of control on the geometries of microminiature devices like the microstripline, whose characteristics vary so greatly with geometry, size, and purity of material. With the realization of such advances must come a sound theoretical knowledge of these devices. This paper is a study of the microstripline and its parameters based upon sound theoretical principles. 2 II. HISTORICAL REVIEW A. Literature Review The microstrip transmission line and its history are often confused with that of the symmetrical or closed stripline, but distinct evidence of work in the area can 1 be found as far back as the 1930's . However, the idea remained reasonably obscure until the early 1950's2 ' 3 ' 4 , when work in the gigahertz region was being expanded. In an attempt to introduce the user to the various strip- line configurations, the IRE prepared a special "Symposium on Microwave Strip Circuits" in 1955 which included some papers cataloging what was then the state of the art in microstrip theory5 ' 6 . In their analysis, Black and Higgins' attempt to use the Schwartz-Christoffel trans- formation was not entirely successful because of their inability to solve some of the key equations in the overall solution. Because of the lack of symmetry in the micro- strip goemetry and because of the apparent mathematical 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. Accordingly, he solved the problem by starting from current equations, assuming that the trans- verse current component was not necessarily zero, and 3 made approximations for a special case and its solution. In spite of this knowledge, most if not all of the subsequent investigators have assumed a TEM mode in their mathematical analyses. Recently, various approaches have been made to circumvent the surprisingly complicated microstrip impedance problem. 9 Kaupp approached the problem by merely assuming a lossless wire-over-ground transmission line using standard TEM transmission line theory and a geometry equivalence to produce reasonably accurate results.