Introduction to Microstrip
Antennas
David R. Jackson
Dept. of ECE
University of Houston
1
David R.
Jackson
Dept. of
ECE
N308 Engineering Building 1
University of
Houston
Houston, TX 77204

4005
Phone: 713

743

4426
Fax: 713

743

4444
Email: djackson@uh.edu
2
Contact Information
Purpose of Short Course
Provide an introduction to microstrip antennas.
Provide a physical and mathematical basis for understanding how
microstrip antennas work.
Provide a physical understanding of the basic physical properties of
microstrip antennas.
Provide an overview of some of the recent advances and trends in
the area (but not an exhaustive survey
–
directed towards
understanding the fundamental principles).
3
Additional Resources
Some basic references are provided at the end of these viewgraphs.
You are welcome to visit a website that goes along with a course at the
University of Houston on microstrip antennas (PowerPoint viewgraphs
from the course may be found there, along with the viewgraphs from this
short course).
4
ECE 6345: Microstrip Antennas
http://www.egr.uh.edu/courses/ece/ece6345/web/welcome.html
Note: You are welcome to use anything that you find on this website,
as long as you please acknowledge the source.
Outline
Overview of microstrip antennas
Feeding methods
Basic principles of operation
General characteristics
CAD Formulas
Radiation pattern
Input Impedance
Circular polarization
Circular patch
Improving bandwidth
Miniaturization
Reducing surface and lateral waves
5
Notation
6
0
k
wavenumber of free space
1
k
wavenumber of substrate
0
intrinsic impedance of free space
1
intrinsic impedance of substrate
r
relative permtitivity (dielectric consta
nt) of substrate
eff
r
effective relative permtitivity
(accouting for fringing of flux lines a
t edges)
eff
rc
complex effective relative permtitivity
(used in the cavity model to account fo
r all losses)
0
wavelength of free space
c
speed of light in free space
0 0 0 0
2/
k
1 0
r
k k
0
0
0
376.7303
0
/
c f
8
2.99792458 10 m/s
c
1 0
/
r
0 0
7
0
12
0
2
0
1
4 10 H/m
1
8.854188 10 F/m
c
c
Outline
Overview of microstrip antennas
Feeding methods
Basic principles of operation
General characteristics
CAD Formulas
Radiation pattern
Input Impedance
Circular polarization
Circular patch
Improving bandwidth
Miniaturization
Reducing surface and lateral waves
7
Overview of Microstrip Antennas
Also called “patch antennas”
One of the most useful antennas at microwave
frequencies (
f
> 1
GHz).
It
usually consists
of a metal “patch” on top of a grounded dielectric substrate.
The patch may be in a variety of shapes, but rectangular and circular are the
most common.
8
Microstrip line feed
Coax feed
Overview of Microstrip Antennas
9
Common Shapes
Rectangular
Square
Circular
Elliptical
Annular ring
Triangular
Invented by Bob Munson in
1972 (but earlier work by
D
echamps
goes back to1953).
Became popular starting in the 1970s.
G.
Deschamps
and W.
Sichak
, “Microstrip Microwave Antennas,”
Proc. of Third
Symp
. on USAF Antenna Research and Development Program,
October 18
–
22, 1953.
R
. E. Munson, “Microstrip Phased Array Antennas,”
Proc. of Twenty

Second
Symp
.
on USAF Antenna Research and Development Program,
October 1972.
R. E. Munson, “Conformal Microstrip Antennas and Microstrip Phased Arrays,”
IEEE
Trans. Antennas
Propagat
., vol. AP

22, no. 1 (January 1974): 74
–
78.
10
Overview of Microstrip Antennas
History
Advantages of Microstrip Antennas
Low profile (can even be “conformal”).
Easy to fabricate (use etching and
photolithography
).
Easy to feed (coaxial cable, microstrip line, etc
.).
Easy to use in an array or incorporate with other microstrip circuit
elements.
Patterns are somewhat hemispherical, with a moderate directivity
(about
6

8
dB
is typical).
11
Overview of Microstrip Antennas
Disadvantages of Microstrip Antennas
Low
bandwidth (but can be improved by a variety
of techniques
). Bandwidths of
a few percent are
typical. Bandwidth is roughly proportional to the substrate
thickness and inversely proportional to the substrate permittivity.
Efficiency
may be lower than with other antennas.
Efficiency
is limited by
conductor and dielectric
losses
*, and by surface

wave loss**.
Only used at microwave frequencies and above (the substrate becomes too
large at lower frequencies).
Cannot handle extremely large amounts of power (dielectric breakdown).
* Conductor and dielectric losses become more severe for thinner
substrates.
** Surface

wave losses become more severe for thicker substrates
(unless air or foam is used).
12
Overview of Microstrip Antennas
Applications
Satellite communications
Microwave communications
Cell phone antennas
GPS antennas
13
Applications include:
Overview of Microstrip Antennas
Microstrip Antenna Integrated into a System: HIC Antenna Base

Station for 28

43 GHz
F
ilter
Diplexer
LNA
PD
K

connector
DC supply Micro

D
connector
M
icrostrip
antenna
Fiber
input with
collimating lens
(Photo courtesy of Dr. Rodney B. Waterhouse)
14
Overview of Microstrip Antennas
Overview of Microstrip Antennas
15
Arrays
Linear array (1

D corporate feed)
2
2 array
2

D 8X8 corporate

fed array
4
8 corporate

fed / series

fed array
Wraparound Array (conformal)
(
Photo
courtesy of Dr. Rodney B. Waterhouse)
16
Overview of Microstrip Antennas
The substrate is so thin that it can be bent to “conform” to the surface.
x
y
h
L
W
Note:
L
is the resonant dimension.
The
width
W
is usually chosen to be larger than
L
(to get higher bandwidth).
However, usually
W
<
2
L
(to avoid problems with the
(0,2)
mode).
r
17
Overview of Microstrip Antennas
Rectangular patch
W
=
1.5
L
is typical.
Circular Patch
x
y
h
a
r
18
Overview of Microstrip Antennas
The location of the feed determines the direction of current flow and hence
the polarization of the radiated field.
Outline
Overview of microstrip antennas
Feeding methods
Basic principles of operation
General characteristics
CAD Formulas
Radiation pattern
Input Impedance
Circular polarization
Circular patch
Improving bandwidth
Miniaturization
Reducing surface and lateral waves
19
Feeding Methods
Some of the more common methods for
feeding microstrip antennas are shown.
20
The feeding methods are illustrated for a rectangular patch,
but the principles apply for circular and other shapes as well.
Coaxial
F
eed
A feed along the centerline
is the most
common
(minimizes higher

order
modes and
cross

pol
).
x
y
L
W
Feed
at (
x
0
,
y
0
)
Surface
current
21
x
r
h
z
Feeding Methods
22
Advantages:
Simple
Directly compatible with coaxial cables
Easy to obtain input
match by adjusting feed position
Disadvantages:
Significant
probe (feed)
radiation for thicker
substrates
Significant probe inductance for thicker substrates
Not easily compatible with arrays
Coaxial
F
eed
2
0
cos
edge
x
R R
L
x
r
h
z
Feeding Methods
x
y
L
W
0 0
,
x y
Advantages:
Simple
Allows for planar
feeding
Easy to use with arrays
Easy to obtain input match
Disadvantages:
Significant line radiation for thicker substrates
For deep notches,
patch current and radiation pattern
may show
distortion
23
Inset Feed
Microstrip line
Feeding Methods
Recent work has shown
that the resonant input
resistance varies as
2
0
2
cos
2
in
x
R A B
L
The coefficients
A
and
B
depend on the notch width
S
but (to a good
approximation) not on the line width
W
f
.
Y. Hu, D. R. Jackson, J. T. Williams, and S. A. Long, “Characterization of the Input Impedance of
the Inset

Fed Rectangular Microstrip Antenna,”
IEEE Trans. Antennas and Propagation
, Vol. 56,
No. 10, pp. 3314

3318, Oct. 2008.
24
L
W
W
f
S
x
0
Feeding Methods
Inset
F
eed
Advantages:
Allows for planar feeding
Less line radiation compared to
microstrip feed
Disadvantages:
Requires multilayer fabrication
Alignment is important for input match
Patch
Microstrip
line
25
Feeding Methods
Proximity

coupled Feed
(Electromagnetically

coupled Feed)
Advantages:
Allows for planar feeding
Can allow for a match
even with
high edge impedances, where
a notch might be too
large (e.g., when using high permittivity)
Disadvantages:
Requires accurate gap fabrication
Requires full

wave design
Patch
Microstrip
line
Gap
26
Feeding Methods
Gap

coupled Feed
Advantages:
Allows for planar
feeding
Feed

line radiation is isolated from patch
radiation
Higher
bandwidth is possible
since probe inductance
is
eliminated (allowing for a thick substrate),
and
also a
double

resonance can be
created
Allows for use of different substrates to optimize antenna
and feed

circuit
performance
Disadvantages:
Requires multilayer fabrication
Alignment is important for input match
Patch
Microstrip
line
Slot
27
Feeding Methods
Aperture

coupled Patch (ACP)
Outline
Overview of microstrip antennas
Feeding methods
Basic principles of operation
General characteristics
CAD Formulas
Radiation pattern
Input Impedance
Circular polarization
Circular patch
Improving bandwidth
Miniaturization
Reducing surface and lateral waves
28
Basic Principles of Operation
The
basic principles are illustrated here for a rectangular patch, but the
principles apply similarly for other patch shapes.
We use the
cavity model
to explain the operation of the patch antenna.
29
Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and Experiment on Microstrip Antennas,”
IEEE Trans. Antennas Propagat
., vol. AP

27, no. 3 (March 1979): 137
–
145.
ˆ
n
h
PMC
z
Basic Principles of Operation
The patch acts approximately as a
resonant cavity
(with short

circuit
(PEC)
walls on top and bottom, open

circuit (PMC) walls on the
edges).
In a cavity, only certain modes are allowed to exist, at different
resonance
frequencies.
If the antenna is excited at a
resonance frequency
, a strong field is set up inside
the cavity, and a strong current on the (bottom) surface of the patch. This
produces significant radiation (a good antenna).
Note: As the substrate thickness gets smaller the patch current radiates less, due to
image cancellation. However, the
Q
of the resonant mode also increases, making the
patch currents stronger at resonance. These two effects cancel, allowing the patch to
radiate well even for small substrate thicknesses.
30
Main Ideas:
Basic
Principles of Operation
As the substrate gets thinner the patch current radiates less, due to image
cancellation.
However, the
Q
of the resonant cavity mode also increases, making the
patch currents stronger at resonance.
These two effects cancel, allowing the patch to radiate well even for thin
substrates.
31
A microstrip antenna can radiate well, even with a thin substrate.
x
r
h
s
J
z
On patch and ground
plane:
0
t
E
ˆ
z
E z E
Inside the patch cavity, because of the thin substrate,
the
electric field vector is approximately independent of
z
.
Hence
ˆ
,,,
z
E x y z z E x y
32
h
,
z
E x y
z
Basic Principles of Operation
Thin Substrate Approximation
1
1
ˆ
,
1
ˆ
,
z
z
H E
j
zE x y
j
z E x y
j
Magnetic field inside patch cavity:
33
Basic Principles of Operation
Thin Substrate Approximation
1
ˆ
,,
z
H x y z E x y
j
Note: The magnetic field is purely horizontal.
(The mode is
TM
z
.)
34
,
H x y
h
,
z
E x y
z
Basic Principles of Operation
Thin Substrate Approximation
On the
edges
of
the patch:
ˆ
0
s
J n
ˆ
0
bot
s
J n
0
bot
t
H
x
y
ˆ
n
L
W
s
J
ˆ
t
On the bottom
surface of
the patch
conductor, at the edge of the patch,
we have
(
J
s
is the sum of the top and bottom surface currents.)
35
bot top
s s
J J
ˆ
bot
s
J z H
Also,
h
0
bot
t
H
Basic Principles of Operation
Magnetic

wall Approximation
0 ( )
t
H
PMC
Since the magnetic field is approximately
independent of
z
, we have an approximate PMC
condition on the entire vertical edge.
ˆ
n
h
PMC Model
36
ˆ
,0
n H x y
or
PMC
h
0
edge
t
H
Actual patch
Basic Principles of Operation
Magnetic

wall Approximation
x
y
ˆ
n
L
W
s
J
ˆ
t
Hence,
0
z
E
n
1
ˆ
,,
z
H x y z E x y
j
ˆ
,0
n H x y
ˆ
ˆ
,0
z
n z E x y
ˆ
ˆ
,0
z
z n E x y
ˆ ˆ ˆ
ˆ ˆ ˆ
,,,
z z z
n z E x y z n E x y E x y n z
37
ˆ
n
h
PMC
(Neumann B.C.)
Basic Principles of Operation
Magnetic

wall Approximation
x
y
ˆ
n
L
W
s
J
ˆ
t
2 2
0
z z
E k E
cos cos
z
m x n y
E
L W
2 2
2
1
0
z
m n
k E
L W
Hence
2 2
2
1
0
m n
k
L W
From
separation of variables
:
(
TM
mn
mode)
x
y
L
W
PMC
,
z
E x y
38
Basic Principles of Operation
Resonance Frequencies
We then have
1 0
r
k k k
2 2
2
1
m n
k
L W
1 0 0 0
r r
k k
Recall that
2
f
Hence
2 2
2
r
c m n
f
L W
0 0
1/
c
39
We thus have
x
y
L
W
PMC
,
z
E x y
Basic Principles of Operation
Resonance Frequencies
2 2
2
mn
r
c m n
f
L W
Hence
mn
f f
(resonance frequency of (
m
,
n
) mode)
40
Basic Principles of Operation
Resonance Frequencies
x
y
L
W
PMC
,
z
E x y
This mode is usually used because
the
radiation pattern has a
broadside beam
.
10
1
2
r
c
f
L
cos
z
x
E
L
0
1
ˆ
sin
s
x
J x
j L L
This mode acts as a
wide
dipole
(width
W
)
that has a resonant length of
0.5
guided
wavelengths in the
x
direction.
x
y
L
W
Current
41
Basic Principles of Operation
Dominant (1,0) mode
This is the mode with the lowest resonance frequency.
The resonance frequency is
mainly controlled
by
the
patch
length
L
and the substrate permittivity.
Resonance
Frequency of Dominant Mode
Note: A higher substrate permittivity allows for a smaller antenna
(miniaturization)
–
but
with a lower
bandwidth.
Approximately, (assuming PMC walls)
This
is equivalent to saying that
the length
L
is one

half of a
wavelength in the
dielectric.
0
/2
/2
d
r
L
1
k L
2 2
2
1
m n
k
L W
(
1,0
) mode:
42
Basic Principles of Operation
The
resonance frequency calculation
can be improved by adding
a
“fringing length extension”
L
to each edge of the patch to get
an
“effective length”
L
e
.
10
1
2
e
r
c
f
L
2
e
L L L
y
x
L
L
e
L
L
Note: Some authors use
effective permitt
ivity in this equation.
43
Basic Principles of Operation
Resonance
Frequency of Dominant Mode
Hammerstad formula:
0.3 0.264
/0.412
0.258 0.8
eff
r
eff
r
W
h
L h
W
h
1/2
1 1
1 12
2 2
eff
r r
r
h
W
44
Note:
Even though the
Hammerstad
formula
involves an effective permittivity, we still use
the
actual substrate permittivity
in the
resonance frequency formula.
10
1
2
2
r
c
f
L L
Basic Principles of Operation
Resonance
Frequency of Dominant Mode
Note:
0.5
L h
This is a good “rule of
thumb” to give a quick estimate.
45
Basic Principles of Operation
Resonance
Frequency of Dominant Mode
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
h /
0
0.75
0.8
0.85
0.9
0.95
1
NORMALIZED FREQUENCY
Hammerstad
Measured
W
/
L
= 1.5
r
= 2.2
The resonance frequency has been normalized by the
zero

order value (without fringing):
f
N
=
f
/
f
0
Results: Resonance
Frequency
46
0
/
h
Basic Principles of Operation
Outline
Overview of microstrip antennas
Feeding methods
Basic principles of operation
General characteristics
CAD Formulas
Radiation pattern
Input Impedance
Circular polarization
Circular patch
Improving bandwidth
Miniaturization
Reducing surface and lateral waves
47
General Characteristics
The bandwidth is directly proportional to substrate thickness
h
.
However, if
h
is greater than about
0.05
0
, the probe inductance
(for a
coaxial feed) becomes
large enough so that matching is difficult.
The bandwidth is inversely proportional to
r
(a foam substrate gives a high
bandwidth
).
The bandwidth of a rectangular patch is proportional to the patch width
W
(but remember, we need to keep
W <
2
L
).
Bandwidth
48
49
2 2
2
mn
r
c m n
f
L W
10
1
2
r
c
f
L
02
2
2
r
c
f
W
2
W L
Width Restriction for a Rectangular Patch
f
c
f
10
f
01
f
02
01
1
2
r
c
f
W
W
=
1.5
L
is typical.
02 01
1 1
2
r
c
f f
W L
General Characteristics
Some Bandwidth Observations
For a typical substrate thickness (
h
/
0
= 0.02
), and a typical substrate
permittivity (
r
= 2.2
) the bandwidth is about
3%.
By using a thick foam substrate, bandwidth of about
10%
can be achieved.
By using special feeding techniques (aperture coupling) and stacked
patches, bandwidths of
100%
have been achieved.
50
General Characteristics
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
h /
0
5
10
15
20
25
30
BANDWIDTH (%)
r
2.2
= 10.8
W
/
L
= 1.5
r
= 2.2
or
10.8
Results: Bandwidth
The discrete data points are measured values.
The
solid curves are from a CAD
formula (given later).
51
0
/
h
10.8
r
2.2
General Characteristics
The resonant input resistance is
fairly
independent of the substrate
thickness
h
unless
h
gets small (the
variation is
then mainly
due to
dielectric and conductor loss
).
The resonant input resistance is proportional to
r
.
The resonant input resistance is directly controlled by the location of the
feed
point
(maximum at edges
x
= 0
or
x
=
L
, zero at center of
patch).
Resonant Input Resistance
L
W
(
x
0
,
y
0
)
L
x
y
52
General Characteristics
The
patch is usually fed along the
centerline
(
y
0
=
W
/ 2
)
to maintain symmetry and thus minimize excitation of undesirable
modes
(
which cause cross

pol
).
Desired mode: (1,0)
L
x
W
Feed
:
(
x
0
,
y
0
)
y
53
Resonant Input Resistance
General Characteristics
For a given mode, it can be shown that the resonant input resistance is
proportional to the square of the cavity

mode field at the feed point.
2
0 0
,
in z
R E x y
For (
1
,
0
) mode:
2
0
cos
in
x
R
L
L
x
W
(
x
0
,
y
0
)
y
54
Resonant Input Resistance
General Characteristics
This will be seen from the cavity

model
eigenfunction
analysis later.
Hence, for (1,
0
) mode:
2
0
cos
in edge
x
R R
L
The value of
R
edge
depends strongly on the substrate
permittivity
(it is proportional to the permittivity).
For a typical patch, it
is often in the range of 100

200
Ohms.
55
Resonant Input Resistance
General Characteristics
L
x
W
(
x
0
,
y
0
)
y
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
h /
0
50
100
150
200
INPUT RESISTANCE (
2.2
r
= 10.8
r
= 2.2
or
10.8
W
/
L
= 1.5
x
0
= L/
4
Results: Resonant
Input
R
esistance
The discrete data points are from a CAD
formula (given later.)
L
x
W
(
x
0
,
y
0
)
y
y
0
=
W
/2
56
0
/
h
10.8
r
2.2
General Characteristics
Region where loss is important
Radiation Efficiency
The radiation efficiency is less than
100%
due to
Conductor
loss
Dielectric
loss
Surface

wave excitation
Radiation efficiency is the ratio of power radiated into
space, to the total input power.
r
r
tot
P
e
P
57
General Characteristics
58
Radiation Efficiency
General Characteristics
surface wave
TM
0
cos (
) pattern
x
y
J
s
r r
r
tot r c d sw
P P
e
P P P P P
P
r
=
radiated power
P
tot
=
total input power
P
c
=
power dissipated by conductors
P
d
=
power dissipated by dielectric
P
sw
=
power launched into surface wave
Hence,
59
Radiation Efficiency
General Characteristics
Conductor and dielectric loss is more important for thinner
substrates (the
Q
of the cavity is higher, and thus more seriously affected by loss).
Conductor loss increases with frequency (proportional to
f
1/2
)
due to the
skin effect.
Conductor
loss is usually more important than dielectric
loss for typical
substrate thicknesses and loss tangents.
1 2
s
R
R
s
is the surface resistance of the metal.
The skin depth of the metal is
.
60
0
2
s
R f
Some observations:
Radiation Efficiency
General Characteristics
Surface

wave power is more important for thicker substrates or for
higher

substrate
permittivities. (The surface

wave power can be
minimized by using a foam substrate.)
61
For a
foam substrate
,
a high
radiation efficiency is obtained by making the
substrate thicker (minimizing the conductor and dielectric losses).
There is no
surface

wave power to worry about.
For a
typical substrate
such as
r
= 2.2
, the radiation efficiency is maximum for
h
/
0
0.02
.
Radiation Efficiency
General Characteristics
r
= 2.2
or
10.8
W
/
L
= 1.5
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
h /
0
0
20
40
60
80
100
EFFICIENCY (%)
exact
CAD
Results:
Efficiency
(Conductor
and dielectric losses are
neglected.)
2.2
10.8
Note: CAD plot uses
Pozar
formula (given later).
62
10.8
r
2.2
0
/
h
General Characteristics
0
0.02
0.04
0.06
0.08
0.1
h /
0
0
20
40
60
80
100
EFFICIENCY (%)
= 10.8
2.2
exact
CAD
r
r
= 2.2
or
10.8
W
/
L
= 1.5
63
7
tan 0.001
3.0 10 [S/m]
Results:
Efficiency
(All losses are accounted for.)
0
/
h
10.8
r
2.2
General Characteristics
Note: CAD plot uses
Pozar
formula (given later).
64
General Characteristics
Radiation Pattern
E

plane: co

pol
is
E
H

plane: co

pol
is
E
Note: For radiation patterns, it is usually more
convenient to place the origin at the middle of the patch
(this keeps the formulas as simple as possible).
x
y
L
W
E plane
H plane
Probe
J
s
Comments on radiation patterns:
The E

plane pattern is typically broader than the H

plane pattern.
The
truncation of the ground plane will cause edge diffraction, which
tends to degrade the pattern by introducing:
Rippling
in the forward direction
Back

radiation
65
Pattern
distortion is more severe in the E

plane, due to the angle
dependence of the vertical polarization
E
on the ground plane, as well as
the surface

wave pattern
.
(Both
vary as
cos
(
))
.
General Characteristics
Radiation Patterns
66
x
y
L
W
E plane
H plane
Edge diffraction is the most serious in the E plane.
General Characteristics
Radiation Patterns
Space wave
cos
E
varies as
J
s
90
60
30
0
30
60
90
120
150
180
210
240
40
30
30
20
20
10
10
E

plane pattern
Red:
infinite substrate and ground plane
Blue:
1 meter ground plane
Note: The E

plane pattern
“tucks in” and tends to
zero at the horizon due to
the presence of the infinite
substrate.
67
General Characteristics
Radiation Patterns
Red:
infinite substrate and ground plane
Blue:
1 meter ground plane
90
45
0
45
90
135
180
225
40
30
30
20
20
10
10
68
H

plane
pattern
General Characteristics
Radiation Patterns
Directivity
The directivity is fairly insensitive to the substrate thickness.
The directivity is higher for lower permittivity, because the patch is
larger.
69
General Characteristics
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
h /
0
0
2
4
6
8
10
DIRECTIVITY (dB)
exact
CAD
= 2.2
10.8
r
r
= 2.2
or
10.8
W
/
L
= 1.5
Results: Directivity
70
0
/
h
2.2
r
10.8
General Characteristics
Outline
Overview of microstrip antennas
Feeding methods
Basic principles of operation
General characteristics
CAD Formulas
Radiation pattern
Input Impedance
Circular polarization
Circular patch
Improving bandwidth
Miniaturization
Reducing surface and lateral waves
71
CAD
Formulas
CAD
formulas for the important properties of the
rectangular microstrip antenna will be shown.
72
D
.
R
.
Jackson,
“
Microstrip
Antennas,”
Chapter
7
of
Antenna
Engineering
Handbook
,
J
.
L
.
Volakis,
Editor,
McGraw
Hill,
2007
.
D
.
R
.
Jackson,
S
.
A
.
Long,
J
.
T
.
Williams,
and
V
.
B
.
Davis,
“Computer

Aided
Design
of
Rectangular
Microstrip
Antennas,”
Ch
.
5
of
Advances
in
Microstrip
and
Printed
Antennas
,
K
.
F
.
Lee
and
W
.
Chen,
Eds
.
,
John
Wiley,
1997
.
D
.
R
.
Jackson
and
N
.
G
.
Alexopoulos,
“Simple
Approximate
Formulas
for
Input
Resistance,
Bandwidth,
and
Efficiency
of
a
Resonant
Rectangular
Patch,”
IEEE
Trans
.
Antennas
and
Propagation
,
Vol
.
39
,
pp
.
407

410
,
March
1991
.
Radiation efficiency
Bandwidth (
Q
)
Resonant input resistance
Directivity
0 0 1 0
1 3 1
1
/16/
hed
r
r
ave
hed
s
r
r d
e
e
R
L
e
h pc W h
where
tan
d
loss tangent of substrate
1
2
s
R
surface resistance of metal
73
Note: “
hed
” refers to a unit

amplitude horizontal electric dipole.
CAD
Formulas
Radiation Efficiency
/2
ave patch ground
s s s
R R R
where
Note: “hed” refers to a unit

amplitude horizontal electric dipole.
2
2
0 1
2
0
1
80
hed
sp
P k h c
1
1
hed
sp
hed
r
hed
hed hed
sw
sp sw
hed
sp
P
e
P
P P
P
3
3
3
0
2
0
1 1
60 1
hed
sw
r
P k h
74
CAD
Formulas
Radiation Efficiency
Note: When we say “unit amplitude” here, we assume peak (not RMS) values.
3
0
1
1
3 1 1
1 1
4
hed
r
r
e
k h
c
Hence,
we have
Physically
, this term is the radiation efficiency of a
horizontal
electric dipole (
hed
) on top of the substrate
.
75
CAD
Formulas
Radiation Efficiency
1
2
1 2/5
1
r r
c
2 4 2
2
2
0 2 4 0 2 0
2 2
2 2 0 0
3 1
1 2
10 560 5
1
70
a
p k W a a k W c k L
a c k W k L
The constants are defined as
2
0.16605
a
4
0.00761
a
2
0.0914153
c
76
CAD
Formulas
Radiation Efficiency
Improved
formula for HED surface

wave power
(due to
Pozar
)
3/2
2
2
0
0 0
2 2
1 0 0 1
1
8
1 ( ) 1 1
r
hed
sw
r r
x
k
P
x k h x x
2
0
1
2
0
1
r
x
x
x
2 2 2
0 1 0 1 0
0
2 2
1
2
1
r r r
r
x
77
D. M.
Pozar
, “Rigorous Closed

Form Expressions for the Surface

Wave Loss of Printed
Antennas,”
Electronics Letters,
vol. 26, pp. 954

956, June 1990.
0 0
tan
s k h s
0
1 0
2
0
1
tan
cos
k h s
k h s
s
k h s
1
r
s
CAD
Formulas
Note: The above formula for the surface

wave power is different from that given in
Pozar’s
paper by
a factor of 2, since
Pozar
used RMS instead of peak values.
1
0 0 0
1 1 16 1
/3
2
ave
s
d
hed
r r
R
pc
h W
BW
h L e
BW
is defined from the frequency limits
f
1
and
f
2
at which
SWR
= 2.0
.
2 1
0
f f
BW
f
(multiply by
100
if you want to get %)
78
1
2
Q
BW
CAD
Formulas
Bandwidth
Comments: For a lossless patch, the bandwidth is
approximately proportional to the patch width and
to the substrate thickness. It is inversely
proportional to the substrate permittivity.
/2
ave patch ground
s s s
R R R
79
CAD
Formulas
Q
Components
1 1 1 1 1
d c sp sw
Q Q Q Q Q
1/tan
d
Q
0 0
( )
2
c
ave
s
k h
Q
R
1 0
3 1
16/
r
sp
L
Q
pc W h
1
hed
r
sw sp
hed
r
e
Q Q
e
3
0
1
1
3 1 1
1 1
4
hed
r
r
e
k h
c
The constants
p
and
c
1
were defined previously.
/2
ave patch ground
s s s
R R R
2
0
cos
max
in edge
x
R R R
L
Probe

feed Patch
0
0
1
0 0 0
4
1 16 1
/3
edge
s
d
hed
r r
L h
W
R
R
pc
W h
h L e
80
CAD
Formulas
Resonant Input Resistance
Comments: For a lossless patch, the resonant resistance is approximately
independent of the substrate thickness. It is inversely proportional to the square
of the patch width. It is proportional to the substrate permittivity.
tanc tan/
x x x
where
2
1
2
1 1
3
tanc
tan
r
r
D k h
pc k h
81
CAD
Formulas
Directivity
1 0
r
k k
1
3
D
pc
For thin substrates:
(The directivity is essentially independent of the substrate thickness.)
82
CAD
Formulas
Directivity
Outline
Overview of microstrip antennas
Feeding methods
Basic principles of operation
General characteristics
CAD Formulas
Radiation pattern
Input Impedance
Circular polarization
Circular patch
Improving bandwidth
Miniaturization
Reducing surface and lateral waves
83
Radiation Pattern
84
h
r
x
P
atch
P
robe
C
oax feed
z
Note: The origin is placed at the center of the patch,
at the top of the substrate, for the pattern calculations.
There are two models often used for calculating the radiation pattern:
Electric current model
Magnetic current model
85
h
r
x
P
atch
P
robe
C
oax feed
patch top bot
s s s
J J J
h
r
x
patch
s
J
probe
s
J
Electric current model:
We keep the physical currents flowing on the patch (and feed).
Radiation Pattern
86
h
r
x
P
atch
P
robe
C
oax feed
Magnetic current model:
We apply the equivalence principle and invoke the (approximate) PMC condition
at the edges.
ˆ
e
s
M n E
Radiation Pattern
Equivalence surface
ˆ
ˆ
e
s
e
s
J n H
M n E
h
r
x
e
s
M
e
s
M
87
Theorem
The electric and magnetic models yield identical patterns
at the
resonance frequency
of the cavity mode.
Assumptions:
1)
The electric and magnetic current models are based on the fields
of a single cavity mode, corresponding to an ideal cavity with
PMC walls.
2)
The probe current is neglected in the electric current model.
D
.
R
.
Jackson
and
J
.
T
.
Williams,
“A
Comparison
of
CAD
Models
for
Radiation
from
Rectangular
Microstrip
Patches,”
Intl
.
Journal
of
Microwave
and
Millimeter

Wave
Computer
Aided
Design
,
vol
.
1
,
no
.
2
,
pp
.
236

248
,
April
1991
.
Radiation Pattern
88
Comments on the models
For the electric current model, it is most convenient to assume an infinite
substrate (in order to obtain a closed

form solution).
Reciprocity can be used to calculate the far

field pattern of electric or
magnetic current sources inside of an infinite layered structure.
The substrate can also be neglected to simplify the far

field calculation.
When an infinite substrate is assumed in the far

field pattern always goes to
zero at the horizon.
D
.
R
.
Jackson
and
J
.
T
.
Williams,
“A
Comparison
of
CAD
Models
for
Radiation
from
Rectangular
Microstrip
Patches,”
Intl
.
Journal
of
Microwave
and
Millimeter

Wave
Computer
Aided
Design
,
vol
.
1
,
no
.
2
,
pp
.
236

248
,
April
1991
.
Radiation Pattern
89
Comments on the models
For the rectangular patch, the electric current model is the simplest since
there is only one electric surface current (as opposed to four edges).
For the rectangular patch, the magnetic current model allows us to classify
the “radiating” and “nonradiating” edges.
For the circular patch, the magnetic current model is the simplest since there
is only one edge (but more than one component of electric surface current,
described by Bessel functions).
Radiation Pattern
On the nonradiating edges, the
magnetic currents are in opposite
directions across the centerline (
x
= 0
).
sin
z
x
E
L
ˆ
e
s
M n E
L
x
W
y
“Radiating edges”
“Nonradiating edges”
e
s
M
s
J
10
ˆ
cos
s
x
J x A
L
(The formulas are based
on
the electric
current
model.)
The origin is at the
center of the patch.
L
r
ε
h
I
nfinite ground plane
and substrate
x
The probe is on the
x
axis.
cos
s
πx
ˆ
J x
L
æ ö
÷
ç
=
÷
ç
÷
ç
è ø
y
L
W
E

plane
H

plane
x
(1,0) mode
90
Radiation Pattern
Rectangular Patch
2
2
sin
cos
2
2
(,,),,
2
2
2 2
y
x
hex
i i
y
x
k W
k L
WL
E r E r
k W
k L
0
sin cos
x
k k
0
sin sin
y
k k
The “hex” pattern is for a
horizontal
electric dipole in the
x
direction
,
sitting on top of the substrate.
or
i
The far

field pattern can be determined by reciprocity.
91
x
y
L
W
s
J
Radiation Pattern
D
.
R
.
Jackson
and
J
.
T
.
Williams,
“A
Comparison
of
CAD
Models
for
Radiation
from
Rectangular
Microstrip
Patches,”
Intl
.
Journal
of
Microwave
and
Millimeter

Wave
Computer
Aided
Design
,
vol
.
1
,
no
.
2
,
pp
.
236

248
,
April
1991
.
0
0
,,sin
,,cos
hex
hex
E r E F
E r E G
where
0
0
2tan
1
tan sec
TE
k h N
F
k h N j N
0
0
2tan cos
cos 1
tan cos
TM
r
k h N
G
k h N j
N
2
sin
r
N
0
0
0
4
jk r
j
E e
r
92
Radiation Pattern
Note: To account for lossy substrate, use
1 tan
r rc r
j
Outline
Overview of microstrip antennas
Feeding methods
Basic principles of operation
General characteristics
CAD Formulas
Radiation pattern
Input Impedance
Circular polarization
Circular patch
Improving bandwidth
Miniaturization
Reducing surface and lateral waves
93
Input Impedance
94
Various models have been proposed over the years for calculating the
input impedance of a microstrip patch antenna.
Transmission line model
The first model introduced
Very simple
Cavity model (
eigenfunction
expansion)
Simple yet accurate for thin substrates
Gives physical insight into operation
CAD circuit model
Extremely simple and almost as accurate as the cavity model
Spectral

domain method
More challenging to implement
Accounts rigorously for both radiation and surface

wave excitation
Commercial software
Very accurate
Can be time consuming
95
1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.6
1.61
1.62
20
10
0
10
20
30
40
50
Frequency (GHz)
Z
i
n
(
)
RLC Circuit model of a Microstrip antenna
1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.6
1.61
1.62
20
10
0
10
20
30
40
50
60
Frequency (GHz)
Z
i
n
(
)
Transmission Line model of a Microstrip antenna
l
eff
includes all loses
1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.6
1.61
1.62
20
10
0
10
20
30
40
50
60
Frequency (GHz)
Z
i
n
(
) (Eigenfunction expansion)
Cavity model of a Microstrip antenna
Results for a typical patch show that the first
three methods agree very well, provided the
correct
Q
is used and the probe inductance is
accounted for.
2.2
tan 0.001
1.524 mm
r
h
7
6.255 cm
/ 1.5
3.0 10 S/m
L
W L
0
0
6.255 cm
0
0.635mm
x
y
a
Input Impedance
Comparison of Simplest Models
Circuit model of patch
Transmission line model of patch
Cavity model (
eigenfunction
expansion) of patch
Transmission Line Model
for
Input Impedance
96
The model accounts for the probe feed to improve accuracy.
The model assumes a rectangular patch
(it is difficult to extend to other shapes).
Input Impedance
We think of the patch as a wide transmission line resonator (length
L
).
0 0
0 0
e
e
x x L
y y W
0
eff
e rc
k k
Denote
1
eff
rc r eff
j l
1 1 1 1 1
eff
d c sp sw
l
Q Q Q Q Q
Note:
L
is from
Hammerstad’s
formula
W
is from Wheeler’s formula
Physical patch dimensions (
W
L
)
97
A CAD formula for
Q
has been given earlier.
Input Impedance
L
e
W
e
x
y
0 0
(,)
e e
x y
PMC
s
J
eff
rc
98
ln4
W h
0.262
0.300
0.412
0.258
0.813
eff
r
eff
r
W
h
L h
W
h
(Hammerstad formula)
1 1
1
2 2
1 12
eff
r r
r
h
W
(Wheeler formula)
Commonly used fringing formulas
Input Impedance
99
where
TL
in in p
Z Z jX
0.57722
( )
Euler's constant
0 0
0 0
1/tan
tan
TL TL eff eff e
in in c c
eff eff e
c c e
Y Z jY k x
jY k L x
0
,
eff eff
c c
Z k
0
x
e
x L
p
L
0
e
x x
p p
X L
Z
in
0 0
1/
eff eff
c c
Y Z
0
0
0
2
ln
2
p
r
X k h
k a
Input Impedance
(from a parallel

plate model of probe inductance)
0 0
1
eff eff
c c
eff
e e
rc
h h
Z
W W
CAD Circuit Model for Input Impedance
100
The circuit model discussed assumes a probe feed.
Other circuit models exist for other types of feeds.
Note: The mathematical justification of the CAD circuit model
comes from the cavity

model analysis, discussed later.
Input Impedance
Near the resonance frequency, the patch cavity can be approximately modeled
as
a resonant
RLC
circuit
.
The resistance
R
accounts for radiation and losses.
A probe inductance
L
p
is added in series, to account for the “probe inductance”
of a probe feed.
L
p
R
C
L
Z
in
P
robe
P
atch
cavity
101
Probe

fed Patch
Input Impedance
0
0
1
in p
R
Z j L
f f
jQ
f f
0
R
Q
L
1
2
BW
Q
BW
is defined here by
SWR
<
2.0
.
0 0
1
2
f
LC
102
L
p
R
C
L
Z
in
Input Impedance
in in in
Z R jX
0
max
in in
f f
R R R
R
is
the input resistance at the resonance of the patch
cavity
(the frequency that maximizes
R
in
).
L
p
R
C
L
103
0
f f
max
in
R
2
0
0
1
in
R
R
f f
Q
f f
Input Impedance
0
f f
(resonance of
RLC
circuit)
The
input resistance
is determined once we know four parameters:
104
f
0
: the resonance frequency of the patch cavity
R
: the input resistance at the cavity resonance frequency
f
0
Q
: the quality factor of the patch cavity
L
p
: the probe inductance
L
p
R
(
R
,
f
0
,
Q
)
L
C
Z
in
CAD formulas
for these four
parameters
have been
given earlier.
0
0
1
in p
R
Z j L
f f
jQ
f f
Input Impedance
4
4.5
5
5.5
6
FREQUENCY (GHz)
0
10
20
30
40
50
60
70
80
R
in
(
)
CAD
exact
Results:
Input
Resistance
vs.
Frequency
r
= 2.2
W
/
L
= 1.5
L
= 3.0 cm
Frequency
where
the input resistance
is maximum (
f
0
)
105
Rectangular patch
Input Impedance
Results: Input
Reactance
vs.
Frequency
r
= 2.2
W
/
L
= 1.5
4
4.5
5
5.5
6
FREQUENCY (GHz)
40
20
0
20
40
60
80
X
in
(
)
CAD
exact
L
= 3.0 cm
Frequency
where the input
resistance is maximum (
f
0
)
Frequency
where the
input impedance is real
Shift
due to probe reactance
106
Rectangular patch
Input Impedance
0.577216
0
0
0
2
ln
2
p
r
X k h
k a
(Euler’s constant)
Approximate CAD formula for
probe
(feed)
reactance (in Ohms)
p p
X L
0 0 0
/376.7303
a
=
probe radius
h
=
probe height
This is based on an infinite parallel

plate model.
107
r
h
2
a
Input Impedance
0
0
0
2
ln
2
p
r
X k h
k a
Feed (probe) reactance increases proportionally with substrate
thickness
h
.
Feed reactance increases for smaller probe radius.
108
If the substrate gets too thick, the probe reactance will make it difficult
to get an input match, and the bandwidth will suffer.
(Compensating techniques will be discussed later.)
Important point:
Input Impedance
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
r
0
5
10
15
20
25
30
35
40
X
f
(
)
CAD
exact
Results: Probe
Reactance
(
X
f
=
X
p
=
L
p
)
x
r
= 2 (
x
0
/
L
)

1
The normalized feed location ratio
x
r
is
zero at the center
of the
patch (
x
=
L
/2
),
and is
1.0
at
the patch edge (
x
=
L
).
r
= 2.2
W
/
L
= 1.5
h
= 0.0254
0
a
= 0.5 m
m
109
r
x
Center
Edge
Rectangular patch
Input Impedance
L
W
(
x
0
,
y
0
)
L
x
y
Cavity Model
110
It is a very efficient method for calculating the input impedance.
It gives a lot of physical insight into the operation of the patch.
The method is extendable to other patch shapes.
Input Impedance
Here we use the cavity model to solve for the
input impedance of the rectangular patch antenna.
Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and Experiment on Microstrip Antennas,”
IEEE Trans. Antennas Propagat
., vol. AP

27, no. 3, pp. 137

145, March 1979.
0 0
0 0
e
e
x x L
y y W
0
eff
e rc
k k
Denote
1
eff
rc r eff
j l
1 1 1 1
eff
d c sp sw
l
Q Q Q Q
Note:
L
is from
Hammerstad’s
formula
W
is from Wheeler’s formula
L
e
W
e
x
y
0 0
(,)
e e
x y
PMC
Physical patch dimensions (
W
L
)
111
Input Impedance
A CAD formula for
Q
has been given earlier.
eff
rc
112
ln4
W h
0.262
0.300
0.412
0.258
0.813
eff
r
eff
r
W
h
L h
W
h
(Hammerstad formula)
1 1
1
2 2
1 12
eff
r r
r
h
W
(Wheeler formula)
Commonly used fringing formulas
Input Impedance
Next, we derive the Helmholtz equation for
E
z
.
Substituting Faradays law into Ampere’s law, we have:
i eff
c
H J j E
E j H
2
2 2
2 2
1
i eff
c
i
e
i
e
i
e
E J j E
j
E j J k E
E E j J k E
E k E j J
113
Input Impedance
(Ampere’s law)
(Faraday’s law)
Hence
Denote
2 2
i
z e z z
E k E j J
(,) (,)
z
x y E x y
where
(,) (,)
i
z
f x y j J x y
Then
114
2 2
(,)
e
k f x y
(scalar Helmholtz equation)
Input Impedance
Introduce “
eigenfunctions
”
mn
(
x
,
y
)
:
For rectangular patch we have, from separation of variables,
2 2
(,) (,)
mn mn mn
x y x y
0
mn
C
n
2 2
2
(,) cos cos
mn
e e
mn
e e
m x n y
x y
L W
m n
L W
115
Input Impedance
Assume an “
eigenfunction
expansion”:
Hence
,
(,) (,)
mn mn
m n
x y A x y
2 2
(,)
e
k f x y
2 2
,,
(,)
mn mn e mn mn
m n m n
A k A f x y
Using the properties of the eigenfunctions, we have
2 2
,
(,) (,)
mn e mn mn
m n
A k x y f x y
This must satisfy
116
Input Impedance
Note that the eigenfunctions are orthogonal, so that
Denote
Next, we multiply by and integrate.
(,)
m n
x y
(,) (,) 0,(,) (,)
mn mn
S
x y x y dS m n m n
2
,(,)
mn mn mn
S
x y dS
2 2
,,
mn e mn mn mn mn
A k f
We then have
117
Input Impedance
Hence, we have
For the patch problem we then have
2 2
,
1
,
mn
mn
mn mn e mn
f
A
k
2 2
,
1
,
i
z mn
mn
mn mn e mn
J
A j
k
The field inside the patch cavity is then given by
,
(,) (,)
z mn mn
m n
E x y A x y
118
Input Impedance
*
*
*
,
*
,
1
(,)
2
1
(,)
2
1
2
1
,
2
i
in z z
V
i
z z
S
i
mn mn z
m n
S
i
mn mn z
m n
P E x y J dV
h E x y J dS
h A J dS
h A J
To calculate the input impedance, we first calculate the complex
power going into the patch as
119
Input Impedance
*
,
*
2 2
,
2
2 2
,
1
,
2
1,1
,
2,
,
1 1
2,
i
in mn mn z
m n
i
i
mn z
mn z
m n
mn mn e mn
i
mn z
m n
mn mn e mn
P h A J
J
h j J
k
J
h j
k
Hence
Also,
2
1
2
in in in
P Z I
so
2
2
in
in
in
P
Z
I
120
Input Impedance
Hence we have
where
2
2
2 2
,
,
1 1
,
i
mn z
in
m n
mn mn e mn
in
J
Z j h
k
I
,0 0
m n m n
121
Input Impedance
Rectangular patch:
where
2 2
2
0
cos cos
mn
e e
mn
e e
eff
e rc
m x n y
L W
m n
L W
k k
1
eff
rc r eff
j l
2 2
0 0
,cos cos
e e
L W
mn mn
e e
m x n y
dx dy
L W
122
Input Impedance
so
To calculate , assume a strip model as shown below
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