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Masters Theses Student Theses and Dissertations

1970

A study of microstrip transmission line parameters utilizing image theory

Joseph Louis Van Meter

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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. 10 11 Wheeler ' approached the problem with a novel use of the Schwartz-Christoffel transformation from which he produced very good results. In fact, due to the scarcity of good experimental data, most subsequent theoreticians compare their results with Wheeler rather than experimental values. Stinehelfer12 utilized finite differences with relaxation to solve the boundary value differential equations for the microstrip geometry in a numerical calculation of the potential and impedance. Yarnash1ta. 13,14 use d a var1a . t' 1ona 1 t ec h n1que . f or the solution of the impedance problem. Suspect in the latter work is the assumption that the charge distribu- tion should take on that of a thin conductor in free space15 with no other materials in its proximity. 4

One of the latest attacks on the microstrip impedance problem was made by Farrer and Adams 16 , who used the method of moments 17 to ascertain the potential distribution.

B. State of the Art

Because of continuing interest, work is moving ever forward in the broad microstrip research area. Roughly three sub-areas of research interest can be defined. These are: (1) Application of microstrips or microstrip- like configurations as devices, (2) Application of micro- strips as an inherent part of a total system, and (3) Fur- ther research on the basic microstrip parameters. Devices which can be inserted in the circuit by merely changing the strip geometry have inspired investi- gators by their simplicity in construction. For example, coupled pairs of microstriplines produce the effect of directional coupling, bends change the VSWR, and tapers produce the equivalent of a transformer in a circuit. Many more equivalents are becoming available. In view of their desirability, these are now being studied in '118,19,20 d e t a~ • On the other hand, a second group of investigators has felt that the state of the art has progressed to such an extent that solid state components and microstrip- lines could be mated to produce completely operational circuits with practical functions. Several of these have . h 1' 21,22,23 recently been note d ~n t e ~terature • 5

A third group of investigators believes that the basic structure and its parameters need more investigation. Some are merely applying new techniques to the impedance 16 24 problem ' . Others are investigating the effects of such changes as substitution of ferrites or semiconductors in place of the dielectric as the base material25 , 26 . The latter works are revealing that devices such as microstrip circulators will be available to miniaturization conscious microwave engineers in the near future. c. Object of Investigation

The objective of this investigation is tq determine the electrical parameters of the microstrip utilizing the image technique. It includes an initial determination of the electric field and the potential function, then a determination of the capacitance and impedance of the line utilizing a first approximation to solve the problem with no initial knowledge of the charge distribution across the strip. With the potential function established it is shown to be possible to determine the charge distribution for a recalculation of the microstrip parameters for higher order approximations. 6

III. THEORY

A. Determination of the Electric Field and Potential ~ the Image TechnTque-

1. The Line Charge in Front of a Ground Plane

Frequently, the concept of potential determination through the image technique is introduced by citing the classical problem of a line charge in front of an infinite ground plane. Figure 1 illustrates this proposition. The line charge has a uniform charge distribution of pL coulombs per meter, is of infinitesimal diameter and is infinite in length. The ground plane is of infinitesimal thickness, infinite in size and has an infinite conductivity. The intervening space is filled with a homogeneous dielectric extending to infinity in all directions. Not apparent in this problem is an induced charge which is distributed on the top of the ground plane and is a direct consequence of the original line charge pL. This also contributes to the overall electric field and potential above the ground plane. Thomson27 considered this problem and theorized that one or more fictitious charges could be placed on the lower side of the plane. These so-called image 7

y HOMOGENEOUS x~z MEDIUM - e:

Figure 1. Filamentary Line Charge Above an Infinite Ground Plane

IMAGE PLANE

p

•\ -- \ ------\r r' - \ -pL -- \ ------\PL yl IMAGE CHARGE ORIGINAL CHARGE

X Figure 2. Profile View of the Ground Plane and Line Charge With the Image Line Charge in Place 8 charges would thus produce the same electric field and potential as that which would have been produced by the induced charges residing on the surface of the image plane. The plane could then be removed and the problem would be reduced to that of a number of finite charges in space. Thomson's postulate may be utilized in this problem and in any problem where the following constraints are met:

1. The potential ~ must satisfy the Laplace

equation v 2 ~ = 0 except at the location of the original charge or charges.

2. ~ = 0 over the plane Y = 0.

3. ~ = 0 at r 1 4. ~ = pL/2~£ ln r as r ~ 0, that is, as P ~ Y1 . For the configuration illustrated in Figures 1 and 2, the function ~ = pL/2n£ ln ! - pL/2~£ ln ;, which is merely the potential of the two line charges, satisfies all four conditions 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.

2. The Potential for a Line Charge in Front of a Dielectric Slab

The problem of a line charge placed in front of an infinite dielectric slab may also be treated by using 9

the image technique. This proposition is illustrated in Figures 3 and 4.

Point P is an arbitrary point of interest at which the potential is desired. Let Y2 ' be the image of Y2 in the plane face created by the dielectric boundary and rand r' be the distances of point P from points Y2 and Y2 ' respectively. Assume that the line charge PL is placed at Y2 and the field in the region Y > 0 is given by placing at point Y2 ' a line charge -pL' such that the potential is given by27

Now suppose that the field inside the dielectric medium is due to a line charge PL" placed at Y2 . Then 1 the potential P "/2TI£ ln satisfies Laplace's* ¢ 2 = L 0 r equation in the dielectric. The values of P L ' and P L " are determined by utilizing the boundary conditions at the interface: 1. The potential is continuous at the interface. 2. The normal component of the displacement vector D is continuous at the interface. Substitution of the potentials into the equations formed by the boundary conditions yields

*The matter of satisfying Laplace's equation is sig~ificant because, for the microstrip, it implies the assumpt~on of a TEM mode. 10

VACUUM

z

. ne Chargeb~·n Front of a semi­ Figure 3. Filam7ntaDielectr1cInfin1te ry L1 . Sla

I '

p / y2 ~ L ORIGINAL ·_jX CHARGE

y . z VACUUM 11

p -pl=p" L L L ( 1)

and

( 2) £0Ely = £ o £ r E2 y.

or

( 3}

Thus,

(4)

Solving Equations (1) and (4} simultaneously yields

1 - £ p I = ( r) ( 5) L - 1 + £ r

and

( 6)

More generally, for a two dielectric system £1 and £:2 1

- £1 - £2 PL I = ( - £ ) PL ( 7) £1 + 2

and

2£1 II = ( ) . ( 8) PL £1 + £2 PL 12

3. Application of the Image Technique to the Micro­ strip for Determining the Potential and Electric Field

a. Determination of the Charge Configuration

The microstrip problem is attacked by considering an axial element of the surface of the upper conductor as a filament of charge (see Figure 5). A sufficient number of image line charges of this filamentary charge satisfy- ing all of the boundary conditions, including those at the air-dielectric interface and at the ground plane, is deter- mined. From this known charge configuration the potential and electric field at arbitrary points, both in and above the dielectric, is found. Finally, the electric field resulting from the total strip is determined by integration about the periphery of the upper conductor, which effect- ively sums the contributions of all of the charge filaments making up its surface. Assumptions are as follows: 1. To simplify the calculations, the dielectric constant of air is considered to be equal to Eo- 2. The extent of the dielectric in both the positive and negative X directions is considered infinite. 3. The ground plane is of zero thickness and infinite in extent in X. 4. Attenuation is excluded from this analysis. 13

UPPER CONDUCTOR AIR

z GROUND PLANE X AIR Figure 5. A Profile View of the Microstrip Showing Point P Which Represents an Axial Filamentary Charge on the Surface of the Upper Conductor

Figure 6. The First Stage of the Image Solution Showing the Fictitious Dielectric and the Image of PL in the Air-Dielectric Boundary 14

Referring to Figure 5, an arbitrary point P on the upper conductor surface, located in terms of X, y and z coordinates, is chosen for consideration. Assume that the point P represents a line charge PL parallel to both the ground plane and the air-dielectric interface. For all first approximations in the following work, the charge distribution about the periphery of the strip is considered constant. Approaching the microstrip problem as a combination of the situations encountered in Sections III-A-1 and III-A-2, the image charge configuration is thus determined. Refer­ ring to Figure 6, the ground plane is considered Y = 0, with the real dielectric extending a distance Y = a. The addition of a fictitious dielectric extending a distance Y = a in the negative Y direction symrneterizes the problem without affecting its final solution. The line charge PL lies a distance b above the dielectric or (a + b) above the ground plane.

Applying the image technique for ¢ above the dielec­ tric results in a first image -pL• at a distance (a- b) above the ground plane. This image line charge results from the consideration of the line charge PL in front of the dielectric at Y = a, as in Section III-A-2. On the other hand, a consideration of the image of PL in the ground plane, as in Section III-A-1, results in a second image line charge -PLat a distance (a+ b) below the ground plane, 15 as shown in Figure 7. Carrying the treatment yet one step further calls for the reflection of the line charge

-PL (itself an image) in the dielectric boundary at Y = +a to form an image line charge PL' at a distance of (3a +b) above the ground plane. This process of reflection and re-reflection would seem to cascade to include an infinite number of line charges both above and below the ground plane. However, the charge PL' at Y = (3a +b) above the ground plane completes the solution, since the charge configuration shown in Figure 8 satisfies all boundary conditions of the problem. Removing the dielectric and the ground plane leaves only the line charges whose potential is given by classical electrostatic theory27 as

P L x2 + (Y_ +_a + b) 2 =~ln[2 .. 2 £ 0 X + (Y - a - b) (9) P ' x2 2 L ln [ + (Y - 3a - b) 4rr£0 x2 + (Y- a+ b) 2 for Y > a. For the potential inside the dielectric region above the ground plane (0 ~ Y < a) , the charge configuration is as shown in Figure 9. Again, the dielectric and ground plane may be removed, leaving only the two line charges PL" and -PL", 16

Y = a+ b . p AIR T L y - a

Y = -a IMAGE OF ORIGINAL CHARGE IN GROUND AIR -p PLANE L y = -a- b

Figure 7. Illustration of the Addition of the Image in the Ground Plane of the Original Line Charge PL 17

..0 + ItS M

Figure 8. The Final Charge Configuration with the Third Image Charge Placed at Y = 3a + b 18

AIR

AIR -p " ! L

Figure 9. The ChargInsidee Conf" th~gu7ation D1electric for the Potential 19 whose potential is given by electrostatic theory27 as

PL" x2 + ( Y + a + b) 2 = ln r·~-__..:...;;;.__~-~~ 0 y < a • (10) 4 '1T£0 x2 + (Y - a- b) 2 J ' <

Thus, the potential for all points above the ground plane has been determined for an arbitrary axial filament of charge on the surface of the rnicrostrip conductor. Hereafter, subscripts 1 and 2 will designate regions above and inside the dielectric respectively.

b. Determination of the Values of pL' and pL"

The values of pL' and pL" are determined by utilizing the boundary conditions at the air-dielectric interface. These are given by the expressions:

Cl

The partial derivatives of ¢ 1 and ¢ 2 with respect to Y are given by 20

Y + a + b Y - a - b [ 2 2 X + (Y + a + b)

[ Y - 3a - b Y - a + b ] x2 + (Y- 3a- b) 2 x2 + (Y - a+ b) 2

( 13) and

= Y + a + b Y - a - b [ x2 + ( Y + a + b) 2 (14)

Substituting Equations (13) and (14) into Equation (ll) and Equations (9) and (10) into Equation (12) yields the following equations:

p + p I = E: p II (15) L L r L and (16)

Solving Equations ( 15) and ( 16) simultaneously gives the

I following expressions for PL and PL II • 1 - E: I r) (17) PL = -(1 E: PL + r and

2 II = ) . (18} PL (1 + E:r PL 21

The potentials can now be written as

PL x2 + (Y + a + b)2 cpl = 4n£ ln [ 2 0 X + (Y - a - b)2

1 - £ r) PL x2 + (Y - 3a - b) 2 + (1 + 4n £ ln ] y > a £r 0 x2 + (Y - a + b)2 ' (19)

and

2 PL x2 + (Y b) 2 ln + a + y 4>2 = 1 + 471'£0 [ 2 ] 0 < < a . £r X + (Y - a - b)2 ' - (20)

c. The Electric Field of the Filamentary Charge in the Region Above the Ground Plane

The electric field above the ground plane is deter- mined by utilizing the expression

E = -'V

(22)

First it is convenient to make a change of variables which will later allow an integration about the conductor surface to determine the total electric field. Letting 22

X= (X- X') and b = Y', the expressions for the potentials become:

{X- X') 2 + (Y + Y' + a) 2 <1>1 = {1n [ J {X- X') 2 + (Y- Y' - a) 2

1 - £ 2 r) (X - X I) 2 + (Y - Y' 3a) ] +

( 23) and

2 PL [ (X - X I) 2 + (Y + Y' + a)2 ¢2 = (1 + ln J . Er 4'TT£0 (X - X I) 2 + (Y - Y' - a)2

( 24)

The partial derivatives of ¢1 and ¢2 with respect to X and Y are given by the expressions:

a¢1 PL - = { [ X X' ax 27T£0 (X - X I ) 2 + (Y + Y' + a)2

X - X' J (X - X I) 2 + (Y - Y' - a)2

1 - £ X - X' + (1 r) [ + Er (X - X I) 2 + (Y - Y' - 3a) 2

X - X' ]} ( 2 5) (X- X') 2 + (Y + Y' - a) 2 23

a

y - Y' a - J (X - X I) 2 + (Y - yl - a)2

1 - e: r) y - Y' - 3a + (1 + e: [ r (X - X I) 2 + (Y - Y' - 3a) 2

y + Y' a - ]} ( 26) (X - X I) 2 + {Y + Y' - a)2 '

a

X - X' J (27) (X - X I) 2 + (Y - Y' - a)2 ' and

a

Y - Y' - a 2 2 J ( 2 8) (X- X') + (Y- Y' - a)

The electric fields in regions 1 and 2 (above and within the dielectric respectively) due to a filamentary line charge are given by substituting Equations {25) through (28) into Equations (21) through (24), and become 24

{[ X - X' (X- X') 2 + (Y + Y' + a) 2

X - X' J (X - X') 2 + (Y - Y' - a) 2

X - X' ]} (X - X') 2 + (Y + Y' - a)2

y + Y' a + a {[ + y (X - X')2 + (Y + Y' + a)2

y - Y' - a J (X - X I) 2 + (Y - Y' - a)2

1 e: - r) y - Y' - 3a +

Y + Y' - a ]} (29) (X- X') 2 + (Y + Y' - a) 2 and 25

a [ x - x• X (X- X') 2 + (Y + Y' + a) 2

X - X' 2 2] (X- X') + (Y- Y' - a)

a y + Y' + a + y [ (X - X I) 2 + (Y + Y' + a)2

y - Y' a - J , (30) (X - X I) 2 + (Y - Y' - a)2 where the constants A and B are defined by

PL 1 PL A = and B = ( 31) 2TT£O 1 + £ r TT£0

d. Determination of the Electric Field of the Upper Conductor in the Region Above the Ground Plane

Referring to Figure 10, the primed coordinate locates the filamentary charge; the unprimed coordinates refer to the arbitrary point R at which the electric field is being determined. The width and thickness of the strip are defined as ~ and t respectively. c The total electric field contributed by all of the filamentary charges making up the surface of the conductor is determined by summing all of their individual vectorial contributions. This operation is achieved by integrating the filamentary field expressions in the primed coordinate 26.

y'= 0

Figure 10. Illustration of the Primed Coordinates and the Dimensions and Boundaries of the Upper Conductor (Points R and P Lie in the Plane of the Page) 27 system about the conductor boundary. A general expression of this operation is given by

Y'=t c dE(X' = ~) + 2 Y'=OI x·r/2 + dE(Y' = tc) X'=-£/2

Y'=t c -9., + dE(X' = 2} (32) I Y'=O or

ET = EA + EB + Ec + ED (33} where the subscripted fields are the individual surface contributions defined by the integrals in Equation (32}. Use of the incremental charge concept gives rise to the relabeling of the electric field in expressions (29) and (30} from E1 and E2 to dE1 and dE2 respectively. Substitution of dE1 and dE2 into Equation (32} gives the contributions of each of the four conductor sides to the total electric field. These are as follows: 28

In region 1,

E1 A x·r/2 (( = a X - X' A X (X - X I) 2 + (Y - a)2 X'=-.R-/2

X - X' J 2 2 (X - X') + (Y + a)

[ X - X' (X- X') 2 + (Y - a) 2

X - x• J} dX' (X - X I) 2 + (Y - 3a) 2

X'=.R-/2

{ [ y - a + a y (X - X I) 2 + (Y - a)2 X'=-.R-/2f

Y + a 2 ] (X- X') 2 + (Y +a)

1 - £ y - a ( r) [ + 1 + £r (X- X') 2 + (Y- a) 2

Y - 3a ] } dX' , (34) (X- X')2 + (Y- 3a)2 29

Y'=t R. E1 X B - 2 p:- = {[ ax (X R. ) 2 a)2 r - 7 + (Y - Y' - Y'=O

X - R. 2 J (X - !) 2 (Y a)2 2 + + Y' +

R. 1 - e:r X - 2 + (1 e: ) [ + r (X - ~)2 Y' a)2 2 + (Y + -

R. X- 2 ]} dY' (X - !)2 + (Y- y' - 3a) 2 2

Y - Y' - a Y't { [ 2 (X - i> 2 + ( Y - Y' - a) Y'=O

y + Y' + a J (X ~) 2 (Y + Y' a)2 - 2 + +

1 - e: y + Y' - a + (1 r) [ + e: (X ~) 2 (Y + Y' a)2 r - 2 + -

y - Y' - 3a 2 ]} dY' ( 35) (X - 2i)2 + (Y _ y• - 3a) 30

x·r/2 {[ x - x• a X (X- X')2 + (Y- tc X'=-R./2

x - x• ------~---~------~] (X- X') 2 + (Y + tc + a) 2

1 - £r X - X' + ( 1 + £ r) [ (X - X' ) 2 + ( Y + t c - a) 2

X - x• ]} dX' X I) 2 (Y - t 3a) 2 (X - + c

X'=R./2 y - tc - a a + y I {[ (X- X') 2 + (Y - t a)2 X'=-R./2 c

y + t c + a 2 J X I) 2 (Y a) (X - + + t c +

1 - £ y + t - a r) c + (1 + [ 2 £r (X - X I) 2 + (Y + tc - a)

y - t - 3a c ]} dX' I ( 36) X I) 2 3a) 2 (X - + (Y - t c - and 31

X + ! a Y't {[ 2 X (X+ ~) 2 + (Y- Y' - a) 2 Y'=O

X + ! 2 J (X + + Y' + a) 2

X + 5I, 2 ]} dY' (X+ ~) 2 + (Y- Y' - 3a) 2

Y'=t c y - Y' a + a { [ - y 51,)2 a)2 f (X + 2 + (Y - Y' - Y'=O

y + Y' + a J !) 2 + (Y + Y' + a)2 (X + 2

1 - E: r) [ y + Y' - a + (1 + 2 E: (X + !) 2 + (Y + Y' - a) r 2

Y - Y' - 3a ]} dY' • ( 3 7) (X+ !) 2 + (Y- Y' - 3a) 2 2

Similarly, the expressions for region 2 are 32

X'=R./2 X - X' f [ _(_X___ X_'..:.;);. 2,__+..::.:...(_Y__ a_)""'l'r' 2 X'=-R./2

~-X' 2] dX' (X- X') 2 + (Y +a) x·r/2 a [ Y - a + y (X- X') 2 + (Y- a) 2 X'=-R./2

Y + a J dX' , ( 38) (X- X') 2 + (Y + a) 2

R. E2 X B - 2 = a Y't [ B X (X - !)2 + (Y - Y' - a)2 Y'=O 2

R. ----..--..,.r-x_-__::2=------;;-2 J dY ' (X - }> 2 + (Y + Y' +a)

Y - Y' - a + a y [ (X _ !) 2 + ( y _ y, _ a) 2 2

Y + y• + a ] dY' , (39) (X- 2 + (Y + Y' + a) 2 !)2 33

E2 x·r/2 c = a [ X - X' """"13 X (X - X I) 2 + (Y - t a)2 X'=-£./2 c

X - X' 2] dX' (X - X I) 2 + (Y + t + a) c

x·r/2 y - t - a + a c y [ (X - X I) 2 + (Y - t - a)2 X'=-£./2 c

y + t + a c dX' I (40) 2 X I) 2 J (X - + (Y + tc + a) and

Y' =It c [ ---;;;---.o..--x_+_;:::..______""5" a X (X+ }> 2 + (Y- Y' - a) 2 Y'=O

Q, X+ -2 ----r;---.oor--___;::::______? J d y I (X + 2 + (Y + Y' + a) 2 ~)2

y I =ftC [ y - Y' - a ____, + a ---;;.._..,~:..___....:::;_ y (X+ ~) 2 + (Y- Y' - a) 2 Y'=O 2

Y + Y' + a 2 ] dY' • ( 41) (X+ 2 + (Y + Y' +a) !)2 34

Performing the integrations gives rise to the electric field expressions due to the respective sides

of the upper conductor. The contribution of the bottom surface is:

El [(X - !.) 2 (Y + a) 2 ] [ (X !) 2 (Y a)2] A ax 2 + + 2 + - = 2 { ln Ao [(X + !.) 2 (Y a) 2 ] [ (X !) 2 + (Y - a) 2] 2 + + 2

[ (X-!.) 2 (Y-a) 2 ] ln 2 } [ (X+!_) 2 (Y-a) 2 ] 2

~ ~ ~ X-2 X+2 X+2 + ay {Arctan Y + a - Arctan y + a + Arctan y _ a

9- R. x--2 1-E X - 2 r) - Arctan Y _ a + ( ~--+--~ (Arctan y _ Ja 1 Er

9- R. ll, X + 2 X + 2 X - 2 - Arctan Y _ Ja + Arctan y _ a - Arctan Y _ a ) } ,

(42)

where the constants A0 and B0 which appear in this and subsequent expressions are defined by

1 and B = 0 1 + Er

The electric field contributions of the remaining

surfaces are given by: 35

El y + t + B c a y = a {Arctan Arctan + a Ao X R. - R. X - 2 X - 2

y t a - c - y + Arctan + Arctan - a R. R. X - 2 X - 2

Y + t - a - Arctan ----~c~-- +Arctan Y- a)} _R. X R. X 2 - 2

R. 2 2 R, 2 2 a [(X--) + (Y+tc+a) ][(x-2 > + (Y-a) ] + ~ {ln----~2--~----~~2~--~R-~2~------~2-- 2 [(X-~) 2 + (Y+a) ][(X-2) + (Y-tc-a) ]

1-Er [(X-~) 2 + (Y-tc-3a) 2 J[(X-~) 2 + (Y-a) 2 ] + (1+£ ) (ln----~R-~2~------~2=-----~R-~2~------~2- ) } I r [ (X-2 ) + (y-3a) ][ (X-2 ) + (Y-tc -a) ]

( 4 3) 36

[(X-!_) 2 + (Y+tc+a> 2 J[(X+~) 2 + (Y-tc-a) 2 J {1n 2 [(X+~)2 + 2 2 2 2 (Y+tc+a) J[(X-~) + (Y-tc-a) J

2 ~ 2 2 (Y-tc-3a) ] [ (X+-2 ) + (Y+t -a) ] c ) } 2 2 2 (Y-t c -3a) J[(X-!)2 + (Y+t c -a) J

x-2~ X+2~ + ay {Arctan Y + a + tc - Arctan y + a + tc

x-! x+! 2 2 - Arctan ~Y~---a---~t-c + Arctan y _ a _ tc

~ ~ 1-£ x- 2 X+2 + ( r) (Arctan Y - Arctan y _ Ja _ t 1 + £r - 3a - tc c

X + X - 2~ 2~ -Arctan t +Arctan y _a+ tc )} ' Y - a + c (44) and 37

El Y + t + a D - Arctan Y + a ---Ao = ax {Arctan ------c~--1 1 X+~ X+'2

Y - a - t - Arctan c + Arctan Y - a t t X + 2 X + 2

Y - t - 3a c Y - 3a - Arctan 1 X + 1 2 X+ 2

Y + tc - a -Arctan +Arctan Y- a)} X + t X + t 2 2

t 2 2 t 2 2 1-e:r [ (X+2) + (Y-tc -3a) ] [ (x+2) + (Y-a) ] + ( l+e: ) ( ln 1 2 2 t 2 2 ) } • r [(x+2) + (Y-3a) ][(x+2) + (Y-tc-a) ]

( 4 5)

Similarly, in the dielectric, 38

E2 2 2 A a [ex-!> + (Y+a) 2 J[(X+~) 2 + (Y-a) ] X [ln 2 Bo = 2 J [(X+~) 2 + (Y+a) 2J[(x-;> 2 + (Y-a) 2 J

R- R- X - 2 X + + a [Arctan - Arctan 2 y Y + a Y + a

t ~ x- 2 X+- - Arctan Y _ a + Arctan Y _ : ] , ( 46)

E2 y + t + a B c y + a = ax [Arctan - Arctan Bo X - R- t 2 X - 2

y - t a c - y - - Arctan + Arctan a J X - R- R- 2 X - 2

a [ (X-!) 2 + (Y+t +a) 2 J[(x-!> 2 + (Y-a) 2 ] 2 c 2 +-r [ln J I [(X-~)2 + 2 2 + (Y-t -a) 2 ] 2 (Y+a) J[(X-~)2 c

( 4 7)

E2 a [(X-~) 2 + 2 2 + (Y-t -a) 2 ] c 2 (Y+tc+a) J[(X+~) c = ~ [ln J Bo 2 + 2 R- 2 + (Y-t -a) 2 ] [(X+~)2 (Y+tc+a) ][ (X-2) c

x-R- x+R- 2 2 + a [Arctan Y + t + a - Arctan y + t + a y c c

t t X - 2 X + 2 ( 48) - Arctan Y _ tc _ a + Arctan y _ tc _ a J 1 39 and,

Y + tc + a [Arctan - Arctan Y + a X + ~ X + .II. 2 2

Y - tc - a - Arctan + Arctan Y - a X+! .II. 2 X+ 2

a +.,_Y

(49)

Thus, the electric field has been determined for all points in space about the microstrip conductor. In the dielectric it is the sum of expressions (46) through (49) and above the dielectric the electric field is determined by summing expressions (42) through {45).

e. Determination of the Potential at a Point Above the Ground Plane Due to the Conducting Strip

Although the potential is not necessary for the primary objective of this paper, the impedance, it is a valuable quantity in many analyses and comes as a by- product of the present work. The potential at a point due to the total conducting strip may be determined by summing the potential contributions of all of the filamentary line charges making up the 40 periphery of the upper conductor, in a manner similar to that presented in Section III-A-3-d of this paper. Equation (50) is a statement of this operation:

= x·r/2 x·=r2 4> (Y I = f 0) + X'=R./2 .X •. =-R./2

R. Y'=Jtc = !.) = -2) + ~f(X' 2 Y'=O (50)

When evaluating expression (50) in region 1, ~f becomes ~l as given in expression (23), and in region 2, cjlf becomes 4> 2 as given in expression (24). After performing the integrations, the potentials are found to be:

2 Jl, 2 2 [ 2 + (Y-a) 1 [(X--) + (Y-t -a) 1 ln 2 2 c [ 2 + (Y+t +a) 2 1 2 2 c

y + tc + a y + a + Arctan Arctan Jl, - R. X - 2 X - 2

y - t - a y a c Arctan - 1 + Arctan 1 - Jl, X - 2 X - 2 41

(Y+t +a) 2 ] c + (X+~) [ ~ 2 (y-t -a) ] c

Y + tc + a + Arctan - Arctan Y + a R. X+2 X+2

y t a - c - y - a + Arctan - Arctan J X + R. X + 2 2

R. X + R. X - 2 Arctan 2 + (Y+a)[Arctan y + a - y + a

R. R. X - 2 X + 2 + (Y-a)[Arctan y _a- Arctan y _a

R. R. X+2 x-2 + (Y+tc+a)[Arctan Y + t +a- Arctan y + t +a c c 42

y y - t - 3a - 3a Arctan c + Arctan £ - £ X - 2 X - 2

y t - a y - a + c + Arctan Q; - Arctan £ J X - 2 X - 2

£ £ X + 2 X - 2 + (Y-3a) [Arctan Y ·- 3a - Arctan y _ 3a

1 £2 2 £2 2 + ~ ln [(X-~) + (Y-3a) ][(X+2) + (Y-3a) ]]

£ £ x-2 X+2 + (Y-a)[Arctan Y - a -Arctan Y - a

1 R, 2 2 £ 2 2 + 2 1n [ (X+~) + (Y-a) ] [ (X-2 ) + (Y-a) ] ] 43

i i X+2 x-2 + (Y-tc-3a)[Arctan Y _ t _ Ja- Arctan Y _ t _ 3a c c

i 2 X - 2 + (Y-t -3a) ]] + (Y+t -a)[Arctan ~~+~t----- c c Y - a c

X + R.. 2 1 i 2 - Arctan Y + t _ a - 2 ln [ (X-2 ) c

(51) the potential in the region above the dielectric, and

Y + tc + a Y + a + Arctan - Arctan i X + ~ X + 2

Y - t - a + Arctan c Y - a ] - Arctan R.. X + i 2 X+ 2

[(x-!> 2 + (Y-a) 2 J[(x- R..> 2 + (Y-t -a> 2 J i 1 2 2 c + (X-2)[2 ln-[-(-X--~}-)~2--+ __(_Y_+_a_)~2-J-[-(X--~~-)~2-+---(Y_+_t_c_+_a_)~2r-J 44

y + t + a + Arctan c y + a Jl.. - Arctan X X Jl.. - 2 - 2

y t a - c - y - + Arctan - Arctan a J X - Jl.. Jl.. 2 X - 2

X + ~ X - Jl.. 2 2 + (Y+tc+a)[Arctan Y + t +a- Arctan Y + t +a c c

Jl.. Jl.. 2 X+2 X-- + (Y+tc+a) ]] + (Y+a)[Arctan Y +a- Arctan Y +!

1 J/..2 2 J/..2 2 - ~ 1n [(X+2 } + (Y+a) ][(X-~} + (Y+a} ]]

Jl.. Jl.. x-2 X+2 + (Y-a}[Arctan -Arctan y _ Y - a a

1 J/..2 2 J/..2 2 - ~ 1n [ (X-~} + (Y-a} ] [ (X+2") + (Y-a) ] ]

X Jl.. X + ~ - ~ ~ + (Y-t -a)[Arctan Y -Arctan y t c - tc - a - c - a

(52) 45 the potential in the dielectric. With the expressions developed thus far, sufficient information is available for preparation of equipotential sketches and electric field plots for a microstripline having any combination of dielectric thickness, conductor thickness and conductor width.

B. Expressions for Capacitance and Impedance

1. A First Approximation to the Capacitance

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 (42) through (49) would give exact values for the electric field. Since this variation is yet unknown, it is assumed to be constant for all first approximations. In the first approximation to the capacitance and line impedance, the capacitance per unit length is computed from the expression

Q (53) c = -¢ I where Q is the charge per unit length on the strip and ¢ is the potential between the strip and the ground plane. The total charge on the strip is determined by summing all of the elements of charge about its surface perimeter and is expressed by 46

p(y Q = Ps dS = p(Y = a)dX + tf2 = a+ t )dX J c s -!G/2T -!G/2

t t Q, Q, + p(X = -2)dY + p(X = 2) dY. (54) r0 r0

Since the charge distribution is assumed constant, the expression for charge per unit length becomes simply

(55)

As a result of the assumption that the conducting strip is a perfect conductor, the potential at all points along the strip is equal. For simplicity, the potential will be computed between the conducting strip and the ground plane at the point X = 0. The expression for the electric field in the dielectric at X = 0 will become useful in this derivation. It is simply the summation of the ay components of expressions (46) through (49). The expression for potential between two points is

b (56) "'ab-+- = J E•dQ. a

Equation (56) becomes, after substitution of Equations (46) through (49) at X = 0, 47 Y=a ne:o< 1 +e:r) 9-/2 cp = 2 [Arctan + Arctan Y 9-/2 PL y - a - t - a j c Y=O - Arctan 9-/2 9-/2 y + a - Arctan Y + t c + a

Y+t +a Y-t -a c c + 1n +( ) 2 + 1n ( ) 2 v1 t/2 11 + 9-/2

1n J1 + (Y-a)2 - 1n (Y+a)2 Ja - m "1 + m y

dY a y ( 56a) where d£ is a dY. y After integration, the expression becomes

[1 + (2tc/t) 2 ] ]

[1 + (4a/£) 2 ][1 + (2a+t /(9-/2)) 2 ] c

2 2 + £[ 2 ~c(1 + Arccot :c- 1n V1 + ( ~£) 2

- Arccot 4a) + :a (1 - 1n V1 + ,Q,

2a+t ./ 2a+t 2a+t c 2 c + ( t/2 ) (1n V1 + ( 272c) - 1- Arccot £/2 )

2a+t 2t 4a - Arctan y- + Arctan £/2c- Arctan~] . (57) 48

For the strip of zero thickness, expression (57) reduces to

1<1>1 = (58)

Substitution of the results of Equations (55) and (57) into Equation (53) gives an expression for the microstrip capacitance as a function of measurable physical parameters. For the zero thickness conductor, the final expres­ sion for the first approximation to the capacitance becomes

(59)

Figures 11, 12, 13, and 14 are plots of the microstrip capacitance for various relative dielectric constants and conductor width to dielectric thickness ratios. Examina- tion of expression (57) indicates that while the value of the capacitance is affected by the inclusion of t , the c conductor thickness, its effect is not appreciable for small thicknesses.

2. Higher Order Approximations to the Capacitance

Expression (59) is called a first approximation to the microstrip capacitance because it is based upon the assumption that the variation of pL across the conductor 700 ZERO THICKNESS CONDUCTOR APPROXIMATION 600

(/) Q <( ei 500 u.. 0 u.... Q. 400 I 1.1.1 u z ~.... 300 u <( Q. 5 200

Sy: -::: l 100

4 - . 1/a - CONDUCTOR WIDTH-TO-DIELECTRIC-fHICi~ESS-RATIO Figure 11. Microstrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For 1/a Values 1-15 and £r Values 1-10) ~ 1.0 6 ZERO THICKNESS CONDUCTOR APPROXIMATION

5

~ 4 < 0:::

Figure 12. tiicrostrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For ~/a Values 1-15 and Er Values 10-90) U1 0 5 r I I I I I I I I I -G ZERO THICKNESS CONDUCTOR APPROXIMATION

4

(/) Q <( 0:::: <( 3 LL. z0 <( z I w u 2 z <( 1-..... u <( 0... <( u 1

60 70 80 t/a - CONDUCTOR WIDTH-TO-DIELECTRIC THICKNESS RATIO Figure 13. Microstrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For t/a Values 5-100 and £ Values 1-10) r lJ1 1-' 40 ZERO THICKNESS CONDUCTOR APPROXIMATION 35

30

(/) Q

10 20 30 40 50 60 70 80 90 100 1/a - CONDUCTOR WIDTH-TO-DIELECTRIC THICKNESS RATIO Figure 14. Microstrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For 1/a Values 5-100 and Er Values 10-90) U1 1\J 53

•;s con s t an t • Yamas h"t~ a 14 cons~"d ered th~s· problem in his determination and concluded that for the type of variation which exists in the microstrip geometry, the assumption of a constant charge variation has little effect on the impedance (and consequently the capacitance) determination. Because the results of the impedance determination of this paper are acceptable, higher order approximations are considered unnecessary for inclusion in the final capacitance and impedance presentations. If desired, however, higher order capacitance approximations are possible through an iterative process carried out in the following manner: From the determina- tion of the electric field based upon a constant pL variation, the flux density is found. Values of pL(X,Y) determined from the flux density evaluated at the conductor surface are used to obtain a new set of values for the electric field through a reapplication of the image technique presented in Section III-A. When iterations on this process converge, the final potential difference between the ground plane and the conductor and the total charge on the conductor are used to determine the value of the microstrip capacitance.

3. The Impedance Problem

In ideal transmission line theory, the ratio of the potential difference between two conductors to the 54

total current flowing on the conductors for a wave

propagating in either the positive or negative direction is the characteristic impedance z of the line c

z = c:z c = =c- ( 6 0)

1 2 where Z = (~/c:) 1 ( 61)

is the intrinsic impedance of the medium surrounding the

conductors, and C is the electrostatic capacitance. Thus, for a homogeneous dielectric extending to infinity, the characteristic impedance differs from the wave impedance by a capacitance factor which is a function of the geometry only.

In the theoretical microstrip problem the impedance determination is complicated by the multi-dielectric character of the medium in which the wave is traveling. In addition, the practical microstrip dielectric is finite in width, which additionally complicates the determination. Thus, the intrinsic impedance in expression (60) cannot In short, the problem is what value should be used for the dielectric constant in expression (61) when the wave in a microstrip travels in a multi-layered medium.

Wheeler10 encountered this problem in his work and proceeded to develop a "field form factor" and a related effective dielectric constant which he used to adjust 55

his expressions to meet the measured values. Seckelmann28 also became aware of this problem and proceeded to take laboratory measurements in an attempt to verify Wheeler's work.

An examination of impedance calculations determined through the use of the expression

(JJ£ £ )1/2 0 r ( 6 2) c I where (JJ£) 1 / 2 is the speed of light in the dielectric material, indicates that its use by many investigators, including Yamashita, is invalid for the microstrip config- uration. This disagreement is quite evident in Figure 15. Cursory analysis of this and similar plots would perhaps indicate that the analysis presented here is merely off by some constant. Exam1nat1on. . o f t h e d ata presente d b y K a1ser ' zg suggests an approach in terms of the propagation constant. Determination of the characteristic impedance of the micro- strip through the use of the expression

, ( 6 3) where r is the propagation constant, gives rise to data such as that found in Figures 16 and 17. Examination of this data shows very close agreement with Wheeler and good agreement with Kaiser's experimental data. Kaiser gives no data concerning the conductor " 140 t \ \ a = .0625 Er = 2.6 I \ \. 120

(/) RAW ~ 100 t \ ~ .,.. CALCULATED VALUES (EQUATION (62)) ...... ~ UJ u z 80

1 2 3 4 5 6 ~/a - CONDUCTOR WIDTH-TO-DIELECTRIC THICKNESS RATIO Figure 15. A Comparison of Raw Theoretical and Experimental Impedance Values

(JI 0'1 MEASURED PROPAGATION CONSTANT MEASURE~ IMPEDANCE 120 ------WHEELER S MODIFIED SCHWARTZ-CHRISTOFFEL 1------IMPEDANCE BY IMAGE IMPEDANCE I l!) TECHNIQUE ...... en ....I -~ u...... ,5100 0 Q UJ UJ UJ 0 a.. u en ~ 80 Q u. UJ 0 a.. 1- ~ z - 60 UJ u u 0::: l­- UJ en a.. -0::: I ~ 40 1- u z ~ " ~ 0::: DIELECTRIC THICKNESS (a) = .0625 1- ~ en I z u e:r = 2.6 0 u z 7 0.... 1 2 3 4 5 6 1- ~ l!) CONDUCTOR WIDTH-TO-DIELECTRIC THICKNESS RATIO ~a.. 0 0::: .15 .20 .25 .30 .40 .45 a.. ~ - CONDUCTOR WIDTH (INCHES) U1 Figure 16. Theoretical and Experimental Impedance Plots (a = 1/16") -...1 120 I \ ------MEASURED PROPAGATION CONSTANT MEASURED IMPEDANCE -130 100 ... \ - WHEELER'S MODIFIED SCHWARTZ-CHRISTOFFEL ---- IMPEDANCE IMPEDANCE BY IMAGE 1- TECHNIQUE fO :c .....(!) en _J -~ ------LL :c ------hn ...... ,0 soi -, r 0 Q LlJ LlJ UJ c.. u en z 60 <( LL Q 0 UJ 0 1- c.. z ~ UJ .... 40 u u .... 0::: 1- UJ '1"'\.'1'~1 ~,..-rr""\TI""' .,..IITI"'IJ'~Ir""l"'l"' ('!"1\ c.. en..... 0::: UJ 20 t- . ------1- u z <( Er = 2.6 <( 10 1- 0::: en <( z :c 0 u u 1 2 3 4 5 6 7 8 9 10 11 12 z i/a - CONDUCTOR WIDTH-TO-DIELECTRIC THICKNESS RATIO .....0 1- <( (!) .05 .10 .15 .20 .2~ .30 .35 <( i - CONDUCTOR WIDTH (INCHES c.. 0 0::: c.. Figure 17. Theoretical and Experimental Impedance Plots (a = 1/32") V1 (X) 59

thickness or the dielectric width. In the calculations the conductor thickness was considered zero. Analysis of the expression indicates that an impedance difference of no more than one or two ohms could be expected with typical conductor thicknesses. 60

IV. CONCLUSIONS, DISCUSSION AND SUGGESTIONS FOR FURTHER WORK

A. Conclusions and Discussion

The results presented in this paper are in reasonable agreement with the literature and certainly suggest the validity of the image technique.

The assumption of a TEM mode through the application of the Laplace equation in the assumptions necessary for the application of the image technique is regrettable from the standpoint of rigor but is unfortunately necessary. This assumption has precipitated comment from various investigators, among them Dukes 7 , who recognized that, although the mode is not TEM, the assumption of a TEM mode produces valid results. Certainly, its success in this paper does not weaken the case for use in further work. The success of the impedance determination lends credence to the validity of the capacitance determination

(although no data can be found in the literature for substantiation) as well as the potential and electric

field determinations. The effect of geometry and size as well as the dielectric constant upon the propagation constant cannot presently be stated with assurance, since no theoretical 61 work can be found in this area. It is felt, however, that since the propagation constant is dependent upon these parameters, some margin for error exists when using experimental values. Of course, a thorough theoretical determination requires a look also at the propagation constant from a theoretical standpoint.

B. Suggestions for Further Work

Certainly regrettable is the fact that a dearth of experimental work is available for examination. One of the shortcomings of the present bit of work in the literature is that some data has been taken by one investigator with one method and a second bit of data has been taken by another investigator by another method, neither one of which can be compared for accuracy or correctness. It is suggested that a comprehensive exper- imental study be undertaken to determine the capacitance, impedance, and propagation constants of microstriplines for a wide range of relative dielectric constants and

~/a ratios. (To clarify the issue, UMR presently has neither the technology nor the support to fabricate microstriplines with sufficient quality control to do a worthwhile study.) Further theoretical work should be done to determine the propagation constant for waves traveling in the multi­ layered dielectric configuration peculiar to the micro­ strip geometry. 62

If it is felt necessary, the iterative process for upgrading the approximations presented in this paper could be attempted by future investigators. Certainly, the work could not be done in closed form but rather would require numerical techniques. Moving beyond the characteristic impedance problem for the simple microstrip configuration, there exist a number of microstrip configurations which need both theoretical and experimental work, including the coupling problem for parallel strips and the determination of the effects of stubs and tapers in microstrip circuits. 63

VITA

Joseph Louis Van Meter was born on October 28, 1945, in Maplewood, Missouri. He received his primary and secondary education in the schools of the Maplewood­ Richmond Heights school district 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 McDonnell­ Douglas Corporation, St. Louis, Missouri, where he was employed as a member of the Engineering Co-op Program. He received a Bachelor of Science degree in Electrical Engineering from the University of Missouri - Rolla, in Rolla, Missouri, in June 1968. He has been enrolled in the Graduate School of the University of Missouri - Rolla since July 1968. He is a member of Tau Beta Pi, Eta Kappa Nu, Phi

Kappa Phi, and the IEEE. 64

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