a study of microstrip transmission line parameters
TRANSCRIPT
Scholars' Mine Scholars' Mine
Masters Theses Student Theses and Dissertations
1970
A study of microstrip transmission line parameters utilizing image A study of microstrip transmission line parameters utilizing image
theory theory
Joseph Louis Van Meter
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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
1971 T2563 c.2
!J~ £ £~ (Advisor) (/
~d~4'3 pages
~9424?
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.
ii
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.
iii
TABLE OF CONTENTS
ABSTRACT
ACKNOWLEDGEMENT
LIST OF ILLUSTRATIONS
LIST OF SYMBOLS
I. INTRODUCTION
II. HISTORICAL REVIEW
A. Literature Review
B. State of the Art
c. Object of Investigation
III. THEORY
A. Determination of the Electric Field and Potential by the Image Technique
1. The Line Charge in Front of a Ground Plane
2. The Potential for a Line Charge in Front of a Dielectric Slab
3. Application of the Image Technique to the Microstrip for Determination of the Potential and Electric Field
a. Determination of the Charge Configuration
b. Determination of the Values of p • and p "
L L
c. The Electric Field of the Filamentary Charge in the Region Above the Ground Plane
iv
Page
ii
iii
vi
viii
1
2
2
4
5
6
6
6
8
12
12
19
21
VITA
d. Determination of the Electric Field of the Upper Conductor in the Region Above the Ground Plane
e. Determination of the Potential at a Point Above the Ground Plane due to the Conducting
v
Page
25
Strip 39
B. Expressions for Capacitance and Impedance
1. A First Approximation to the
45
Capacitance 45
2. Higher Order Approximations to the Capacitance 48
3. The Impedance Problem 53
IV. CONCLUSIONS, DISCUSSION AND SUGGESTIONS FOR FURTHER WORK
A. Conclusions and Discussion
B. Suggestions for Further Work
60
60
61
63
BIBLIOGRAPHY 64
1
2
3
4
5
6
7
8
9
10
11
LIST OF ILLUSTRATIONS
Filamentary Line Charge Above an Infinite Ground Plane
Profile View of the Ground Plane and Line Charge With the Image Line Charge in Place
Filamentary Line Charge in Front of a Semi-Infinite Dielectric Slab
Profile View of the Dielectric, Interface, and Line Charge With the Image Line Charge in Place
A Profile View of the Microstrip Showing Point P Which Represents an Axial Filamentary Charge on the Surface of the Upper Conductor
The First Stage of the Image Solution Showing the Fictitious Dielectric and the Image of pL in the Air-Dielectric 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 Placed at Y = 3a + b
The Charge Configuration for the Potential Inside the Dielectric
Illustration of the Primed Coordinates and the Dimensions and Boundaries of the Upper Conductor
Microstrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For ~/a values 1-15 and ~r values 1-10)
vi
Page
7
7
10
10
13
13
16
17
18
26
49
Figure
12
13
14
15
16
17
Microstrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For ~/a values 1-15 and £r values 10-90)
Microstrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For ~/a values 5-100 and £r values 1-10)
Microstrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For ~/a values 5-100 and £ values 10-90)
r
A Comparison of Raw Theoretical and Experimental Impedance Values
Theoretical and Experimental Impedance Plots (a = 1/16")
Theoretical and Experimental Impedance Plots (a = 1/32")
vii
Page
50
51
52
56
57
58
A
a
a X
a y
B
b
c
p
Q
r
LIST OF SYMBOLS
pL/2TI€0
The Thickness of the Microstrip Dielectric
A Unit Vector in the X Direction
A Unit Vector in the Y Direction
pL/TI€0(l + €r)
The Height of the Filamentary Charge Above the Air-Dielectric Interface
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
An Arbitrary Point of Interest at Which a Potential or Electric Field Value is Being Determined
The Charge Residing on the Upper Conductor of the Micros trip
The Distance From Point P to pL
viii
r'
X
y
z
z
z c
r
E: r
The Distance From Point P to the Image Line Charge
The Thickness of the Upper Conductor of the Micros trip
The Width Coordinate of the Microstrip
The Height Coordinate on the Microstrip
The Length Coordinate on the Microstrip
Wave Impedance (Bar Distinguishes Between Impedance and Z Coordinate)
Characteristic Impedance of the Microstripline (The Phrase "The Impedance" Unless Otherwise Stated Refers to the Characteristic Impedance) (The Bar Distinguishes from the Z Length Coordinate)
Propagation Constant
An Arbitrary Dielectric Constant
-9 The Permittivity of Free Space - l/36rr x 10 Farad/ Meter
The Relative Permittivity of an Arbitrary Material
The Permittivity in Region 1
s 2 The Permittivity in Region 2
p I
L
p II
L
00
A Line of Filamentary Charge
An Image Filamentary Charge Resulting From the AirDielectric Boundary
An Image Filamentary Charge Resulting From the 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
Infinity
ix
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.
1
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's 2 ' 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 theory 5 ' 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 .
wu 8 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
2
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.
. 13,14 d . t' 1 t h . f Yarnash1ta use a var1a 1ona ec n1que 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.
3
One of the latest attacks on the microstrip impedance
16 problem was made by Farrer and Adams , 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
d t '118,19,20 e 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
d . h 1' 21,22,23 recently been note ~n t e ~terature •
4
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 material 25 , 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.
5
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.
Thomson 27 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
6
y
x~z HOMOGENEOUS MEDIUM - e:
Figure 1. Filamentary Line Charge Above an Infinite Ground Plane
-pL ------IMAGE CHARGE
X
IMAGE PLANE
p • -- \
r' ----- \ ----- \r -- \
\ \PL
yl ORIGINAL CHARGE
Figure 2. Profile View of the Ground Plane and Line Charge With the Image Line Charge in Place
7
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.
4.
~ = 0 at r
~ = pL/2~£ 1 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
8
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
9
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 by 27
<P 1 =P L ln !. 2TI EO r
Now suppose that the field inside the dielectric
medium is due to a line charge PL" placed at Y2 . Then 1
the potential ¢ 2 = P "/2TI£ ln satisfies Laplace's* L 0 r
equation in the dielectric. The values of P ' and P " L 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.
VACUUM
z
a semi·n Front of . ne Chargeb~ ry L1 . Sla Filam7ntaDielectr1c Figure 3. Infin1te
y
I '
/ y2 ~ p L
ORIGINAL ·_jX CHARGE
. z VACUUM
10
p -pl=p" L L L
and
£0El = £ £ E2 y o r y.
or
Thus,
Solving Equations (1) and (4} simultaneously yields
and
More
and
1 - £ p I
L = ( r)
- 1 + £ r
generally, for a two
- £1 - £2 PL
I = ( - £ ) £1 + 2
II ( 2£1
) PL = £1 + £2
dielectric system £1 and
PL
PL .
11
( 1)
( 2)
( 3}
(4)
( 5)
( 6)
£:2 1
( 7)
( 8)
3. Application of the Image Technique to the Microstrip 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-
12
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.
UPPER CONDUCTOR
z X
AIR
GROUND PLANE 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
13
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
14
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 theory 27 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",
Y = a+ b
AIR L T. p
AIR -p L
y = -a- b
y - a
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
16
..0
+ ItS
M
Figure 8. The Final Charge Configuration with the Third Image Charge Placed at Y = 3a + b
17
18
AIR
AIR ! -p " L
Figure 9. The Charg e Conf" Inside th~gu7ation for D1electric the Potential
whose potential is given by electrostatic theory 27 as
PL" x2 + + 2 ( Y a + b) = ln r·~-__..:...;;;.__~-~~ 0 4 '1T£0 x2 + (Y - a- b) 2 J ' <
y < a • (10)
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<Pl £ ra <P 2 aY = aY
Y=a Y=a
and
<Pl = ¢2
Y=a Y=a
The partial derivatives of ¢ 1 and ¢ 2 with respect to Y
are given by
( 11)
(12)
19
20
Y + a + b [ 2 2
X + (Y + a + b)
Y - a - b
[ Y - 3a - b x2 + (Y- 3a- b) 2
Y - a + b ]
x2 + (Y - a+ b) 2
( 13)
and
= [ x2 + ( Y + a + b) 2 Y + a + b Y - a - b
(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
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:
PL I = -(1 r)
PL (17) + E: r
and
II (1
2 ) (18} PL = PL .
+ E:r
The
cpl =
and
4>2 =
potentials can now be written as
PL ln
x2 + (Y + a + b)2 4n£ 0 [ 2
X + (Y - a - b)2
1 - £ PL x2 (Y - 3a - b) 2 + (1 r) ln +
+ £r 4n £0 x2 + b)2 ]
+ (Y - a
2 PL ln
x2 + (Y + a + b) 2 0 [ 2 ] < 1 + 471'£0 ' £r (Y - b)2 -X + a -
c. The Electric Field of the Filamentary Charge in the Region Above the Ground Plane
' y > a
(19)
y < a
(20)
The electric field above the ground plane is deter-
mined by utilizing the expression
E = -'V<jl, ( 21)
where the del operator is defined in cartesian coordinates
in two dimensions by the expression
(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
21
.
X= (X- X') and b = Y', the expressions for the potentials
become:
<1>1 = {X- X') 2 + (Y + Y' + a) 2 {1n [ J
{X- X') 2 + (Y- Y' - a) 2
1 - £ (X - X I) 2 (Y - Y' 2 <r +
r) ln[ + - 3a) ] + X I) 2 a)2 } Er (X - + (Y + Y' -
( 23)
and
¢2 (1 2 PL
ln [ (X - X I) 2 + (Y + Y' + a)2 J =
4'TT£0 X I) 2 + a)2 . + Er (X - (Y - Y' -
( 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 I ) 2 a)2 (X - + (Y + Y' +
X - X' J (X - X I) 2 + (Y - Y' - a)2
1 - £ X X' r) [ -+ (1 + X I) 2 3a) 2 Er (X - + (Y - Y' -
X - X' ]}
(X- X') 2 + (Y + Y' - a) 2 ( 2 5)
22
a<Pl PL { [ y + y I + a w- = 27fe:o X') 2 a)2 (X - + (Y + Y' +
y - Y' - a X I) 2 a)2
J (X - + (Y - yl -
1 - e: + (1 + r) [ y - Y' - 3a
e: (X - X I) 2 (Y 3a) 2 r + - Y' -
y + Y' - a X I) 2 a)2
]} ' ( 26) (X - + {Y + Y' -
a<P2 2 PL [ X - X'
ax = 1 + e: 27fe:o X I) 2 + a)2 r (X - + (Y + Y'
X - X' J (27) X I) 2 + a)2 ' (X - (Y - Y' -
and
a<P2 2 PL [ y + Y' + a
w- = 1 27fe:o X I) 2 a)2 + e:r (X - + (Y + Y' +
Y - Y' - a 2 2 J
(X- X') + (Y- Y' - a) ( 2 8)
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
23
24
{[ X - X' (X- X') 2 + (Y + Y' + a) 2
X - X' J (X - X') 2 + (Y - Y' - a) 2
X - X' ]}
X') 2 a)2 (X - + (Y + Y' -
{[ y + Y' + a + a X')2 + a)2 y (X - + (Y + Y'
y - Y' - a J (X - X I) 2 + (Y - Y' - a)2
1 - e: y Y' 3a r) - -+ <r + [ X') 2 3a) 2 e: (X - + (Y - Y' -r
Y + Y' - a ]}
(X- X') 2 + (Y + Y' - a) 2 (29)
and
where the
a [ x - x• X (X- X') 2 + (Y + Y' + a) 2
(X- X') + (Y- Y' - a)
X - X' 2 2]
+ a [ y + Y' + a
y (X - X I) 2 + (Y + Y' + a)2
y - Y' - a J X I) 2 a)2
, (30) (X - + (Y - Y' -
constants A and B are defined by
A PL
and B 1 PL = 2TT£O = ( 31) 1 + £ TT£0 r
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
25
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
I c
+ dE(X' = ~) 2
Y'=O
+ x·r/2
dE(Y' = tc)
X'=-£/2
Y'=t
I
c
dE(X' -9.,
+ = 2}
Y'=O
or
ET = EA + EB + Ec + ED
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 dE 2 respectively.
Substitution of dE1 and dE 2 into Equation (32}
(32)
(33}
gives the contributions of each of the four conductor sides
to the total electric field. These are as follows:
28
In region 1,
E1 x·r/2 (( X - X' A = a X I) 2 a)2 A X (X - + (Y -
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
f { [ y - a + a
X I) 2 a)2 y (X - + (Y -X'=-.R-/2
Y + a 2 ] (X- X') 2 + (Y +a)
1 - £ y - a ( r) [
+ 1 + £r (X- X') 2 + (Y- a) 2
Y - 3a ] } dX' , (X- X')2 + (Y- 3a)2
(34)
E1 B
p:- =
Y'=t R.
ax r X - 2 {[ R. ) 2 a)2 (X - + (Y - Y' -7
+
+
Y'=O
(X -
1 -(1 +
(X -
X - R. 2 J !) 2
2 + (Y + Y' + a)2
e:r e: ) [
(X -r
R. X- 2
!)2 + (Y-2
X R.
- 2
~)2 + (Y + Y' -2
]} dY' y' - 3a) 2
a)2
Y't Y - Y' - a { [ 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) [ ~) 2 (Y + Y' a)2 e: (X - + -r 2
y - Y' - 3a 2 ]} dY' i)2 + (Y _ y• - 3a) (X - 2
29
( 35)
and
x - x• a X
x·r/2 {[ X'=-R./2
(X- X')2 + (Y- tc
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 - X I) 2 + (Y - t 3a) 2
c
X'=R./2
I {[ y - tc - a + a
(X- X') 2 (Y -y + t c X'=-R./2
y + t + a c 2 J
(X - X I) 2 + (Y + t + a) c
1 - £ y + t - a r) c
+ (1 + [ X I) 2 £r (X - + (Y + tc - a)
y - t - 3a dX' c ]} (X - X I) 2 + (Y - t - 3a) 2
c
30
a)2
2
I ( 36)
a X Y't Y'=O
(X +
X + ! {[ 2
(X+ ~) 2 + (Y- Y' - a) 2
X + ! 2
+ Y' + a) 2 J
X + 5I, 2 ]} dY'
(X+ ~) 2 + (Y- Y' - 3a) 2
Y'=t c y -
f + a { [ - Y' a
51,)2 a)2 y (X + + (Y - Y' -Y'=O 2
y + Y' + a J (X + !) 2 + (Y + Y' + a)2
2
1 - E: y + Y' + (1 +
r) [ - a E: (X + !) 2 (Y + Y' 2
r + - a) 2
Y - Y' - 3a ]} dY' • (X+ !) 2 + (Y- Y' - 3a) 2
2
Similarly, the expressions for region 2 are
31
( 3 7)
E2 B = B
X'=R./2
f X - X' [ _(_X ___ X_'..:.;);.2,__+..::.:...(_Y __ a_)""'l'r'2
X'=-R./2
~-X' 2] dX' (X- X') 2 + (Y +a)
+ a y
x·r/2 [ Y - a
(X- X') 2 + (Y- a) 2
a
X'=-R./2
Y + a J dX' , (X- X') 2 + (Y + a) 2
Y't X R. - 2 [
!)2 X (X - + (Y - Y' -Y'=O 2
R.
----..--..,.r-x_-__::2=--------;;-2 J dY ' }> 2 + (Y + Y' +a) (X -
a)2
+ a y Y - Y' - a
[ (X _ !) 2 + ( y _ y, _ a) 2 2
Y + y• + a ] dY' , (X- !) 2 + (Y + Y' + a) 2
2
32
( 38)
(39)
E2 c
""""13 =
and
X - X' x·r/2
[ a X I) 2 a)2 X (X - + (Y - t c
+
a X
X'=-£./2
X - X' 2] dX'
(X - X I) 2 + (Y + t + a) c
x·r/2 y - t - a [ c a
X I) 2 y (X - + (Y - t X'=-£./2 c
y + t + a dX' c 2 J
(X - X I) 2 + (Y + tc + a)
Y' =It c [ ---;;;---.o..--x_+_;:::..__ ____ ""5"
(X+ }> 2 + (Y- Y' - a) 2 Y'=O
Q, X+ -2
----r;---.oor--___;::::__ _____ ? J d y I
~) 2 + (Y + Y' + a) 2 (X + 2
-
I
+ a y y I =ftC [ ---;;.._..,~:..___....:::;_ ___ _, y - Y' - a
(X+ ~) 2 + (Y- Y' - a) 2 2 Y'=O
Y + Y' + a 2 ] dY' • (X+ !) 2 + (Y + Y' +a)
2
a)2
33
(40)
( 41)
Performing the integrations gives rise to the
electric field expressions due to the respective sides
of the
surface
El A
Ao
upper
is:
ax = 2
conductor.
{ ln [(X - !.) 2
2
[(X + !.) 2 2
[ (X-!.) 2 ln 2
[ (X+!_) 2 2
The contribution of
+ (Y + a) 2 ] [ (X + !) 2 2
+ (Y + a) 2 ] [ (X !) 2 2
the bottom
+ (Y -
+ (Y -
(Y-a) 2 ]
(Y-a) 2 ]
a)2]
a) 2]
}
~ ~ ~ X-2 X+2 X+2
+ ay {Arctan Y + a - Arctan y + a + Arctan y _ a
9- R. x--2 1-E ( r) - Arctan Y _ a + ~1--+--~ Er
X - 2 (Arctan y _ Ja
9- R. X + 2 X + 2
- Arctan Y _ Ja + Arctan y _ a
ll, X - 2
- Arctan Y _ a ) } ,
(42)
where the constants A0 and B0 which appear in this and
subsequent expressions are defined by
and B 0
= 1
1 + Er
The electric field contributions of the remaining
surfaces are given by:
34
35
El y + t + a B {Arctan c y + a Ao
= a - Arctan X
X R. R. - 2 X - 2
y - t - a Arctan c y - a + R. + Arctan R. X - 2 X - 2
- Arctan Y + t - a ----~c~-- +Arctan Y- a)}
X _R. X R. 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-
r [ (X-2 ) + (y-3a) ][ (X-2 ) + (Y-tc -a) ] ) } I
( 4 3)
and
[(X-!_) 2 + {1n 2
[(X+~)2 + 2
(Y+tc+a> 2 J[(X+~) 2 + (Y-tc-a) 2 J
(Y+tc+a) 2 J[(X-~) 2 + (Y-tc-a) 2 J
2 ~ 2 2
36
(Y-tc-3a) ] [ (X+-2 ) + (Y+t -a) ] c ) }
(Y-t -3a) 2 J[(X-!) 2 + (Y+t -a) 2J c 2 c
~ ~ 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
+ ( 1 r) (Arctan Y - Arctan y _ Ja _ t + £r - 3a - tc c
~ X + ~ X - 2 2 -Arctan t +Arctan y _a+ tc )} ' Y - a + c
(44)
Y + t + a El D --- = ax {Arctan Ao ------c~-- - Arctan Y + a 1 1
X+~ X+'2
- Arctan Y - a -
t X + 2
t c + Arctan Y - a
t X + 2
Y - t - 3a c
X + 1 2
Y - 3a - Arctan 1 X+ 2
Y + tc - a -Arctan +Arctan Y- a)}
X + t X + t 2 2
t 2 2 t 2 2
37
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 a [ex-!> 2 + (Y+a) 2 J[(X+~) 2 2 + (Y-a) ] A X [ln 2 Bo = 2
(Y+a) 2J[(x-;> 2 J [(X+~) 2 + + (Y-a) 2 J
X -R-
X + R-
+ a [Arctan 2 - Arctan 2 y Y + a Y + a
t ~ x- 2 X+-- Arctan Y _ a + Arctan Y _ : ] , ( 46)
E2 y + t + a y + B [Arctan c - Arctan a
Bo = ax R- t X - 2 X - 2
y - t - a - Arctan c + Arctan y - a J R- R-X - 2 X - 2
[ (X-!) 2 (Y+t +a) 2 J[(x-!> 2 2 a + + (Y-a) ] +-r [ln 2 c 2 J [(X-~)2 (Y+a) 2 J[(X-~) 2 2 I
+ + (Y-t -a) ] 2 2 c
( 4 7)
E2 [(X-~) 2 (Y+tc+a) 2 J[(X+~) 2 2 a + + (Y-t -a) ] c ~ [ln 2 c J = [(X+~)2 2 R- 2 2 Bo 2 + (Y+tc+a) ][ (X-2) + (Y-t -a) ] 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
- Arctan Y _ tc _ a + Arctan y _ tc _ a J 1 ( 48)
and,
Y + tc + a [Arctan - Arctan Y + a
X + ~ X + .II. 2 2
Y - tc - a - Arctan
X+! 2
+ Arctan Y - a .II.
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.
39
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
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'=R./2
4> (Y I = f 0) +
x·=r2 .X •. =- R./2
R. Y'=Jtc = -2) + ~f(X'
Y'=O
= !.) 2
(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:
y
+ Arctan
y
+ Arctan
[ <x-l> 2 + ln 2
[ <x-l) 2 + 2
+ tc + a -Jl,
X - 2
- t -c a -1
X - 2
2 Jl, 2 (Y-a) 1 [(X--) + 2
(Y+a) 2 1 [ex-!> 2 + 2
y + a Arctan R.
X - 2
y - a 1 Arctan Jl,
X - 2
2 (Y-t -a) 1 c
(Y+t +a) 2 1 c
40
+ (X+~) [ ~
Y + tc + a + Arctan - Arctan Y + a
R. X+2 X+2
y - t - a + Arctan c y - a - Arctan R.
X + 2 X + 2
X + R. X R. 2 - 2
+ (Y+a)[Arctan - Arctan y + a y + a
R. R. X - 2 X + 2
+ (Y-a)[Arctan y _a- Arctan y _a
J
(Y+t +a) 2 ] c 2 (y-t -a) ] c
R. R. X+2 x-2
+ (Y+tc+a)[Arctan Y + t +a- Arctan y + t +a c c
41
42
y 3a y - t - 3a - c
+ Arctan £ - Arctan X
£ - 2 X - 2
y + t - a y - a c J + Arctan Q; - Arctan £ 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 -Arctan Y - a Y - a
1 R, 2 2 £ 2 2 + 2 1n [ (X+~) + (Y-a) ] [ (X-2 ) + (Y-a) ] ]
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
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 X + i
2
Y - a ] - Arctan R..
X+ 2
(51)
[(x-!> 2 + (Y-a) 2 J[(x- 2R..> 2 + (Y-t -a> 2 J i 1 2 c
+ (X-2)[2 ln-[-(-X--~}-)~2--+ __ (_Y_+_a_)~2-J-[-(X--~~-)~2-+---(Y_+_t_c_+_a_)~2r-J
43
44
y + t + a + Arctan c Arctan
y + a Jl.. -
X X Jl.. - 2 - 2
y - t - a + Arctan c y - a J - Arctan Jl.. Jl.. X - 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 c = - I ¢ (53)
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
Q = J
Ps dS = T p(Y = a)dX + tf2 p(y = a+ t )dX c
s -!G/2 -!G/2
t t
+ r p(X Q, r Q, = -2)dY + p(X = 2) dY. (54)
0 0
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
-+- = J E•dQ. "'ab
a
(56)
Equation (56) becomes, after substitution of Equations (46)
through (49) at X = 0,
46
47 Y=a
ne:o< 1 +e:r) cp 2 j [Arctan 9-/2 9-/2 = + Arctan Y
PL y - a - t - a c
Y=O
- Arctan 9-/2 - Arctan Y 9-/2
y + a + t + a c
v1 Y+t +a
) 2 11 + Y-t -a
) 2 + 1n +( c + 1n ( c t/2 9-/2
J1 + - 1n (Y-a)2 - 1n "1 + (Y+a)2 Ja m m y
a dY ( 56a) y
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~c(1 + Arccot 2:c- 1n V1 + ( 2 ~£) 2
+ :a ( 1 - 1n V 1 + - Arccot 4 a) ,Q,
2a+t c + ( t/2 )
./ 2a+t 2 (1n V 1 + ( 272 c) - 1- Arccot
2a+t c £/2 )
4a - Arctan y- + Arctan 2a+t 2t
£/2c- Arctan~] . (57)
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
48
700
(/) Q <(
600
ei 500 u.. 0 u .... Q. 400 I
1.1.1 u z ~ 300 .... u <( Q.
5 200
100
ZERO THICKNESS CONDUCTOR APPROXIMATION
4 - .
Sy: -::: l
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
5
~ 4 < 0::: < u. 0
~ 3 z I
L1J u ~ 2 .... -u < 0.. < ull-
ZERO THICKNESS CONDUCTOR APPROXIMATION
'l"~~~ ~ €".
~ 10
Figure 12. tiicrostrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For ~/a Values 1-15 and Er Values 10-90)
U1 0
(/) Q <( 0:::: <( LL. 0 z <( z I
w u z <( 1-..... u <( 0... <( u
5 r I I I I I I I I I -G
4
3
2
1
ZERO THICKNESS CONDUCTOR APPROXIMATION
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
35
30
(/) Q
25 <C a:: <C u. 0 z 20 <C z I
UJ 15 u z <C 1-.... u 10 <( a. <( u
5
Figure 14.
ZERO THICKNESS CONDUCTOR APPROXIMATION
10 20 30 40 50 60 70 80 90 100 1/a - CONDUCTOR WIDTH-TO-DIELECTRIC THICKNESS RATIO
Microstrip Capacitance Versus Conductor Width-to-Dielectric Thickness Ratio (For 1/a Values 5-100 and Er Values 10-90)
U1 1\J
;s con t t Y h"t 14 "d · • s an • amas ~ a cons~ 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
53
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
where Z = (~/c:) 11 2
= c:z =c- ( 6 0)
( 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.
Wheeler 10 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
54
his expressions to meet the measured values. Seckelmann 28
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
c I
where (JJ£) 1 / 2 is the speed of light in the dielectric
( 6 2)
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.
. . f h d d b K ' zg Exam1nat1on o t e ata presente y a1ser
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
55
" 140 t \ \ a = .0625
Er = 2.6 I \ \.
120
(/)
~ 100 t \ ~ RAW .,.. CALCULATED VALUES (EQUATION (62)) . ... ....... ~
UJ u 80 z <t Q UJ tl..
/ :::E - 60 u ..... 1- MEASURED (/) - 40 VALUES 0::: UJ 1-u <t 0::: 20 <( ::r: u
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
-en ~
120
5100 .......,
UJ u ~ 80 Q UJ a.. ~
- 60 u -len -0:::
~ 40 u ~ 0::: ~ I u
--------------MEASURED PROPAGATION CONSTANT MEASURE~ IMPEDANCE WHEELER S MODIFIED SCHWARTZ-CHRISTOFFEL IMPEDANCE BY IMAGE IMPEDANCE
TECHNIQUE
" DIELECTRIC THICKNESS (a) = .0625 e:r = 2.6
1
Figure 16.
2 3 4 5 6
CONDUCTOR WIDTH-TO-DIELECTRIC THICKNESS
.15 .20 .25 .30 ~ - CONDUCTOR WIDTH (INCHES)
Theoretical and Experimental Impedance Plots
7
RATIO
.40 .45 (a = 1/16")
1-I l!) ...... ....I
u. 0
Q UJ
0 UJ a.. en u. 0
1-z UJ u 0::: UJ a.. I
1-z ~ 1-en z 0 u z 0 .... 1-~ l!) ~ a.. 0 0::: a..
U1 -...1
120 I \
----------- MEASURED PROPAGATION CONSTANT -130 MEASURED IMPEDANCE 100 ... \ - WHEELER'S MODIFIED SCHWARTZ-CHRISTOFFEL ---- IMPEDANCE BY IMAGE IMPEDANCE fO 1-TECHNIQUE :c
(!) -soi
..... -, r en ---------------------- _J
~ :c -------------------------------------------- hn
LL 0 0 ........,
Q LlJ LlJ
UJ c.. u 60 en z <( LL Q 0 UJ 0 1-c.. ~ 40
z .... UJ u
u 0::: ....
1- UJ en '1"'\.'1'~1 ~,..-rr""\TI""' .,..IITI"'IJ'~Ir""l"'l"' ('!"1\ c.. ..... 0::: 20 UJ . -----t- - 1-u Er = 2.6 z <( 10 <(
0::: 1-<( en :c z u 0
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:::
Figure 17. Theoretical and Experimental Impedance Plots (a = 1/32") c.. V1 (X)
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.
59
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
60
from the standpoint of rigor but is unfortunately necessary.
This assumption has precipitated comment from various
7 investigators, among them Dukes , 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
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.
61
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.
62
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.
63
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