Kolias, N.J., Compton, R.C., Fitch, J.P., Pozar, D.M. “Antennas”
The Electrical Engineering Handbook
Ed. Richard C. Dorf
Boca Raton: CRC Press LLC, 2000
38
Antennas
38.1Wire
Short Dipole?Directivity?Magnetic Dipole?Input
Impedance?Arbitrary Wire Antennas?Resonant Half-Wavelength
Antenna?End Loading?Arrays of Wire Antennas?Analysis of
General Arrays?Arrays of Identical Elements?Equally Spaced
Linear Arrays?Planar (2-D) Arrays?Yagi–Uda Arrays?Log-
Periodic Dipole Arrays
38.2Aperture
The Oscillator or Discrete Radiator?Synthetic
Apertures?Geometric Designs?Continuous Current Distributions
(Fourier Transform)?Antenna Parameters
38.3Microstrip Antennas
Introduction?Basic Microstrip Antenna Element?Feeding
Techniques for Microstrip Antennas?Microstrip Antenna
Arrays?Computer-Aided Design for Microstrip Antennas
38.1 Wire
N.J. Kolias and R.C. Compton
Antennas have been widely used in communication systems since the early 1900s. Over this span of time
scientists and engineers have developed a vast number of different antennas. The radiative properties of each
of these antennas are described by an antenna pattern. This is a plot, as a function of direction, of the power
P
r
per unit solid angle W radiated by the antenna. The antenna pattern, also called the radiation pattern, is
usually plotted in spherical coordinates q and j. Often two orthogonal cross sections are plotted, one where
the E-field lies in the plane of the slice (called the E-plane) and one where the H-field lies in the plane of the
slice (called the H-plane).
Short Dipole
Antenna patterns for a short dipole are plotted in Fig. 38.1. In these plots the radial distance from the origin
to the curve is proportional to the radiated power. Antenna plots are usually either on linear scales or decibel
scales (10 log power).
The antenna pattern for a short dipole may be determined by first calculating the vector potential A [Collin,
1985; Balanis, 1982; Harrington, 1961; Lorrain and Corson, 1970]. Using Collin’s notation, the vector potential
in spherical coordinates is given by
(38.1)Aaa=-
-
m
p
qq
q0
0
4
Idl
e
r
jkr
r
( cos sin)
N.J. Kolias
Raytheon Company
R.C. Compton
Cornell University
J. Patrick Fitch
Lawrence Livermore Laboratory
David M. Pozar
University of Massachusetts
at Amherst
? 2000 by CRC Press LLC
where k
0
= 2p/l
0
, and I is the current, assumed uniform, in the short dipole of length dl (dl << l
0
). Here the
assumed time dependence e
jw t
has not been explicitly shown. The electric and magnetic fields may then be
determined using
(38.2)
The radiated fields are obtained by calculating these fields in the so-called far-field region where r >> l.
Doing this for the short dipole yields
(38.3)
where Z
0
= . The average radiated power per unit solid angle W can then be found to be
(38.4)
FIGURE 38.1 Radiation pattern for a short dipole of length dl (dl << l
0
). These are plots of power density on linear scales.
(a) E-plane; (b) H-plane; (c) three-dimensional view with cutout.
EA
A
HA=- +
?? ×
=?′j
j
w
wm e m
00 0
1
Ea
Ha
=
=
-
-
jZ Idlk
e
r
jIdlk
e
r
jk r
jk r
00
0
0
0
4
4
sin
sin
q
p
q
p
q
j
m
0
e
0
¤
D
DW
P
rIZdlk
r
r
(, )
{}()
sinqj q
p
=?′×=
1
2
32
22
0
2
0
2
2
2
e EH*a **
? 2000 by CRC Press LLC
Directivity
The directivity D(q,j) and gain G(q,j) of an antenna are defined as
(38.5)
Antenna efficiency, h, is given by
(38.6)
For many antennas h ?1 and so the words gain and directivity can be used interchangeably. For the short dipole
(38.7)
The maximum directivity of the short dipole is 3/2. This single number is often abbreviated as the antenna
directivity. By comparison, for an imaginary isotropic antenna which radiates equally in all directions,
D(q,j) = 1. The product of the maximum directivity with the total radiated power is called the effective isotropic
radiated power (EIRP). It is the total radiated power that would be required for an isotropic radiator to produce
the same signal as the original antenna in the direction of maximum directivity.
Magnetic Dipole
A small loop of current produces a magnetic dipole. The far fields for the magnetic dipole are dual to those of
the electric dipole. They have the same angular dependence as the fields of the electric dipole, but the polar-
ization orientations of E and H are interchanged.
(38.8)
where M = p r
0
2
I for a loop with radius r
0
and uniform current I.
Input Impedance
At a given frequency the impedance at the feedpoint of an antenna can be represented as Z
a
= R
a
+ jX
a
. The
real part of Z
a
(known as the input resistance) corresponds to radiated fields plus losses, while the imaginary
part (known as the input reactance) arises from stored evanescent fields. The radiation resistance is obtained
from R
a
= 2P
r
/|I|
2
where P
r
is the total radiated power and I is the input current at the antenna terminals. For
electrically small electric and magnetic dipoles with uniform currents
D
P
P
G
P
P
r
r
r
(,)
(,)
(,)
(,)
qj
p
qj
p
qj
p
qj
p
==
Radiated power per solid angle
Total radiated power/4
/
/
Radiated power per solid angle
Total input power/4
/
/
in
DDW
DDW
4
4
h
qj
qj
o=
P
P
G
D
r
in
(,)
(,)
D(,) sinqj q=
3
2
2
Ha
Ea
=-
=
-
-
Mk
e
r
MZk
e
r
jkr
jkr
0
2
00
2
0
0
4
4
sin
sin
q
p
q
p
q
j
? 2000 by CRC Press LLC
(38.9)
The reactive component of Z
a
can be determined from X
a
= 4w(W
m
-W
e
)/|I|
2
where W
m
is the average magnetic
energy and W
e
is the average electric energy stored in the near-zone evanescent fields. The reflection coefficient,
G, of the antenna is just
(38.10)
where Z
0
is the characteristic impedance of the system used to measure the reflection coefficient.
Arbitrary Wire Antennas
An arbitrary wire antenna can be considered as a sum of small current dipole elements. The vector potential
for each of these elements can be determined in the same way as for the short dipole. The total vector potential
is then the sum over all these infinitesimal contributions and the resulting E in the far field can be found to be
(38.11)
where the integral is over the contour C of the wire, a is a unit vector tangential to the wire, and r￠ is the radial
vector to the infinitesimal current element.
Resonant Half-Wavelength Antenna
The resonant half-wavelength antenna (commonly called the half-wave dipole) is used widely in antenna
systems. Factors contributing to its popularity are its well-understood radiation pattern, its simple construction,
its high efficiency, and its capability for easy impedance matching.
The electric and magnetic fields for the half-wave dipole can be calculated by substituting its current
distribution, I = I
0
cos(k
0
z), into Eq. (38.11) to obtain
(38.12)
The total radiated power, P
r
, can be determined from the electric and magnetic fields by integrating the
expression 1/2 Re {E ′ H* · a
r
} over a surface of radius r. Carrying out this integration yields P
r
= 36.565 |I
0
|
2
.
The radiation resistance of the half-wave dipole can then be determined from
R
dl
R
r
a
a
=
?
è
?
?
?
÷
=
?
è
?
?
?
÷
80
320
2
0
2
6 0
0
4
p
l
p
l
electric dipole
magnetic dipole
G=
-
+
ZZ
ZZ
a
a
0
0
Eaa
ar
r
() [( ) ]()rjkZ
e
r
Il e dl
jk r
rr
c
jk
=×-￠￠
-
× ￠
ò
00
0
0
4p
Ea
Ha
=
?
è
?
?
?
÷
=
?
è
?
?
?
÷
-
-
jZ I
e
r
jI
e
r
jk r
jk r
00
0
2
2
2
2
0
0
cos cos
sin
cos cos
sin
p
q
qp
p
q
qp
q
j
? 2000 by CRC Press LLC
(38.13)
This radiation resistance is considerably higher than the radiation resistance of a short dipole. For example, if
we have a dipole of length 0.01l, its radiation resistance will be approximately 0.08 W (from Eq. 38.9). This
resistance is probably comparable to the ohmic resistance of the dipole, thereby resulting in a low efficiency.
The half-wave dipole, having a much higher radiation resistance, will have much higher efficiency. The higher
resistance of the half-wave dipole also makes impedance matching easier.
End Loading
At many frequencies of interest, for example, the broadcast band, a half-wavelength
becomes unreasonably long. Figure 38.2 shows a way of increasing the effective
length of the dipole without making it longer. Here, additional wires have been
added to the ends of the dipoles. These wires increase the end capacitance of the
dipole, thereby increasing the effective electrical length.
Arrays of Wire Antennas
Often it is advantageous to have several antennas operating together in an array.
Arrays of antennas can be made to produce highly directional radiation patterns.
Also, small antennas can be used in an array to obtain the level of performance of
a large antenna at a fraction of the area.
The radiation pattern of an array depends on the number and type of antennas
used, the spacing in the array, and the relative phase and magnitude of the excitation
currents. The ability to control the phase of the exciting currents in each element
of the array allows one to electronically scan the main radiated beam. An array that varies the phases of the
exciting currents to scan the radiation pattern through space is called an electronically scanned phased array.
Phased arrays are used extensively in radar applications.
Analysis of General Arrays
To obtain analytical expressions for the radiation fields due to an array one must first look at the fields produced
by a single array element. For an isolated radiating element positioned as in Fig. 38.3, the electric field at a far-
field point P is given by
(38.14)
where K
i
(q,j) is the electric field pattern of the individual element, a
i
e
–jai
is the excitation of the individual
element, R
i
is the position vector from the phase reference point to the element, i
p
is a unit vector pointing
toward the far-field point P, and k
0
is the free space wave vector.
Now, for an array of N of these arbitrary radiating elements the total E-field at position P is given by the
vector sum
(38.15)
This equation may be used to calculate the total field for an array of antennas where the mutual coupling
between the array elements can be neglected. For most practical antennas, however, there is mutual coupling,
R
P
I
a
r
=?
2
73
0
2
**
W
FIGURE 38.2Using end
loading to increase the
effective electrical length of
an electric dipole.
EK
Ri
iii
jk
ae
ip i
=
×-
(,)
[( )]
qj
a
0
EEK
Ri
tot
==
×-
=
-
=
-
??ii
jk
ipi
i
N
i
N
ae(,)
[( )]
qj
a
0
0
1
0
1
? 2000 by CRC Press LLC
and the individual patterns will change when the element is placed in the array. Thus, Eq. (38.15) should be
used with care.
Arrays of Identical Elements
If all the radiating elements of an array are identical, then K
i
(q,j) will be the same for each element and
Eq. (38.15) can be rewritten as
(38.16)
This can also be written as
(38.17)
The function f(q,j) is normally called the array factor or the array polynomial. Thus, one can find E
tot
by just
multiplying the individual element’s electric field pattern, K(q,j), by the array factor, f(q,j). This process is
often referred to as pattern multiplication.
The average radiated power per unit solid angle is proportional to the square of E
tot
. Thus, for an array of
identical elements
(38.18)
Equally Spaced Linear Arrays
An important special case occurs when the array elements are identical and are arranged on a straight line with
equal element spacing, d, as shown in Fig. 38.4. If a linear phase progression, a, is assumed for the excitation
currents of the elements, then the total field at position P in Fig. 38.4 will be
FIGURE 38.3Diagram for determining the far field due to radiation from a single array element. (Source: Reference Data
for Radio Engineers, Indianapolis: Howard W. Sams & Co., 1975, chap. 27–22. With permission.)
EK
Ri
tot
=
×-
=
-
?
(,)
[( )]
qj
a
ae
i
jk
i
N
ip i0
0
1
EK
Ri
tot
where ==
×-
=
-
?
(,)(,) (,)
[( )]
qj qj qj
a
ffae
i
jk
ipi
i
N
0
0
1
D
DW
P
f
r
(,)
~(,)(,)
qj
qj qj** *K
22
? 2000 by CRC Press LLC
(38.19)
where y = k
0
d cos q – a.
Broadside Arrays
Suppose that, in the linear array of Fig. 38.4, all the excitation currents are equal in magnitude and phase (a
0
=
a
1
= . . . = a
N – 1
and a = 0). The array factor, f(y), then becomes
(38.20)
This can be simplified to obtain the normalized form
(38.21)
Note that f'(y) is maximum when y = 0. For our case, with a = 0, we have y = k
0
d cosq. Thus f'(y) will be
maximized when q = p/2. This direction is perpendicular to the axis of the array (see Fig. 38.4), and so the
resulting array is called a broadside array.
FIGURE 38.4 A linear array of equally spaced elements.
EK
KK
tot
=
==
-
=
-
=
-
?
?
(, )
(, ) (, )( )
( cos )
qj
qj qj y
qa
y
ae
ae f
n
jn k d
n
N
n
jn
n
N
0
0
1
0
1
faea
e
e
jn
n
N jN
j
()y
y
y
y
==
-
-
=
-
?0
0
1
0
1
1
￠ ==f
f
aN
N
N
()
()
sin
sin
y
y
y
y
0
2
2
? 2000 by CRC Press LLC
Phased Arrays
By adjusting the phase of the elements of the array it is possible to vary the direction of the maximum of the
array’s radiation pattern. For arrays where all the excitation currents are equal in magnitude but not necessarily
phase, the array factor is a maximum when y = 0. From the definition of y, one can see that at the pattern
maximum
Thus, the direction of the array factor maximum is given by
(38.21b)
Note that if one is able to control the phase delay, a, the direction of the maximum can be scanned without
physically moving the antenna.
Planar (2-D) Arrays
Suppose there are M linear arrays, all identical to the one pictured in Fig. 38.4, lying in the yz-plane with
element spacing d in both the y and the z direction. Using the origin as the phase reference point, the array
factor can be determined to be
(38.22)
where a
y
and a
z
are the phase differences between the adjacent elements in the y and z directions, respectively.
The formula can be derived by considering the 2-D array to be a 1-D array of subarrays, where each subarray
has an antenna pattern given by Eq. (38.19).
If all the elements of the 2-D array have excitation currents equal in magnitude and phase (all the a
mn
are
equal and a
z
= a
y
= 0), then the array will be a broadside array and will have a normalized array factor given by
(38.23)
Yagi–Uda Arrays
The Yagi–Uda array can be found on rooftops all over the world—the standard TV antenna is a Yagi–Uda array.
The Yagi–Uda array avoids the problem of needing to control the feeding currents to all of the array elements
by driving only one element. The other elements in the Yagi–Uda array are excited by near-field coupling from
the driven element.
The basic three-element Yagi–Uda array is shown in Fig. 38.5. The array consists of a driven antenna of
length l
1
, a reflector element of length l
2
, and a director element of length l
3
. Typically, the director element is
shorter than the driven element by 5% or more, while the reflector element is longer than the driven element
by 5% or more [Stutzman and Thiele, 1981]. The radiation pattern for the array in Fig. 38.5 will have a
maximum in the +z direction.
kd
0
cosqa=
q
a
=
?
è
?
?
?
÷
-
cos
1
0
kd
fae
mn
jnkd jmkd
m
M
n
N
zy
(,)
[( cos ) ( sinsin )]
qj
qa qja
=
-+ -
=
-
=
-
??
00
0
1
0
1
￠ =
?
è
?
?
?
÷
?
è
?
?
?
÷
?
è
?
?
?
÷
?
è
?
?
?
÷
f
Nkd
N
kd
Mkd
M
kd
(,)
sin cos
sin cos
sin sinsin
sin sinsin
qj
q
q
qj
qj
0
0
0
0
2
2
2
2
? 2000 by CRC Press LLC
One can increase the gain of the Yagi–Uda array by
adding additional director elements. Adding additional
reflector elements, however, has little effect because the
field behind the first reflector element is small.
Yagi–Uda arrays typically have directivities between 10
and 100, depending on the number of directors [Ramo
et al., 1984]. TV antennas usually have several directors.
Log-Periodic Dipole Arrays
Another variation of wire antenna arrays is the log-peri-
odic dipole array. The log-periodic is popular in applica-
tions that require a broadband, frequency-independent
antenna. An antenna will be independent of frequency if
its dimensions, when measured in wavelengths, remain
constant for all frequencies. If, however, an antenna is designed so that its characteristic dimensions are periodic
with the logarithm of the frequency, and if the characteristic dimensions do not vary too much over a period
of time, then the antenna will be essentially frequency independent. This is the basis for the log-periodic dipole
array, shown in Fig. 38.6.
In Fig 38.6, the ratio of successive element positions equals the ratio of successive dipole lengths. This ratio
is often called the scaling factor of the log-periodic array and is denoted by
(38.24)
Also note that there is a mechanical phase reversal between successive elements in the array caused by the
crossing over of the interconnecting feed lines. This phase reversal is necessary to obtain the proper phasing
between adjacent array elements.
To get an idea of the operating range of the log-periodic antenna, note that for a given frequency within the
operating range of the antenna, there will be one dipole in the array that is half-wave resonant or is nearly so.
This half-wave resonant dipole and its immediate neighbors are called the active region of the log-periodic
array. As the operating frequency changes, the active region shifts to a different part of the log-periodic. Hence,
the frequency range for the log-periodic array is roughly given by the frequencies at which the longest and
shortest dipoles in the array are half-wave resonant (wavelengths such that 2L
N
< l < 2L
1
) [Stutzman and
Thiele, 1981].
FIGURE 38.6The log-periodic dipole array. (Source: D.G. Isbell, “Log periodic dipole arrays,” IRE Transactions on Antennas
and Propagation, vol. AP-8, p. 262, 1960. With permission.)
FIGURE 38.5 Three-element Yagi–Uda antenna.
(Source: Shintaro Uda and Yasuto Mushiake, Yagi–Uda
Antenna, Sendai, Japan: Sasaki Printing and Publishing
Company, 1954, p. 100. With permission.)
t= =
++
z
z
L
L
n
n
n
n
11
? 2000 by CRC Press LLC
Defining Terms
Antenna gain:The ratio of the actual radiated power per solid angle to the radiated power per solid angle
that would result if the total input power were radiated isotropically.
Array: Several antennas arranged together in space and interconnected to produce a desired radiation pattern.
Directivity: The ratio of the actual radiated power per solid angle to the radiated power per solid angle that
would result if the radiated power was radiated isotropically. Oftentimes the word directivity is used to
refer to the maximum directivity.
Phased array:An array in which the phases of the exciting currents are varied to scan the radiation pattern
through space.
Radiation pattern: A plot as a function of direction of the power per unit solid angle radiated in a given
polarization by an antenna. The terms radiation pattern and antenna pattern can be used interchangeably.
Related Topics
37.1 Space Propagation?69.2 Radio
References
C.A. Balanis, Antenna Theory Analysis and Design, New York: Harper and Row, 1982.
R. Carrel, “The design of log-periodic dipole antennas,” IRE International Convention Record (part 1), 1961,
pp. 61–75.
R.E. Collin, Antennas and Radiowave Propagation, New York: McGraw-Hill, 1985.
R.F. Harrington, Time Harmonic Electromagnetic Fields, New York: McGraw-Hill, 1961.
D.E. Isbell, “Log periodic dipole arrays,” IRE Transactions on Antennas and Propagation, vol. AP-8, pp. 260–267,
1960.
P. Lorrain and D.R. Corson, Electromagnetic Fields and Waves, San Francisco: W.H. Freeman, 1970.
S. Ramo, J.R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, New York: John
Wiley & Sons, 1984.
W.L. Stutzman and G.A. Thiele, Antenna Theory and Design, New York: John Wiley & Sons, 1981.
S. Uda and Y. Mushiake, Yagi–Uda Antenna, Sendai, Japan: Sasaki Printing and Publishing Company, 1954.
Further Information
For general-interest articles on antennas the reader is directed to the IEEE Antennas and Propagation Magazine.
In addition to providing up-to-date articles on current issues in the antenna field, this magazine also provides
easy-to-read tutorials. For the latest research advances in the antenna field the reader is referred to the IEEE
Transactions on Antennas and Propagation. In addition, a number of very good textbooks are devoted to antennas.
The books by Collin and by Stutzman and Thiele were especially useful in the preparation of this section.
38.2 Aperture
J. Patrick Fitch
The main purpose of an antenna is to control a wave front at the boundary between two media: a source (or
receiver) and the medium of propagation. The source can be a fiber, cable, waveguide, or other transmission
line. The medium of propagation may be air, vacuum, water, concrete, metal, or tissue, depending on the
application. Antenna aperture design is used in acoustic, optic, and electromagnetic systems for imaging,
communications, radar, and spectroscopy applications.
There are many classes of antennas: wire, horn, slot, notch, reflector, lens, and array, to name a few (see
Fig. 38.7). Within each class is a variety of subclasses. For instance, the horn antenna can be pyramidal or
conical. The horn can also have flaring in only one direction (sectoral horn), asymmetric components, shaped
? 2000 by CRC Press LLC
HIGH-SPEED SPACE DATA
COMMUNICATIONS
? 2000 by CRC Press LLC
edges, or a compound design of sectoral and pyramidal combined. For all antennas, the relevant design and
analysis will depend on antenna aperture size and shape, the center wavelength l, and the distance from the
aperture to a point of interest (the range, R). This section covers discrete oscillators, arrays of oscillators,
synthetic apertures, geometric design, Fourier analysis, and parameters of some typical antennas. The emphasis
is on microwave-type designs.
SI/TelSys Inc., Columbia, Maryland, is a company formed to commercialize NASA high-data-
rate telemetry technology originally developed at Goddard Space Flight Center’s Microelectronic
Systems Branch. Today, TSI/TelSys Inc. designs, manufactures, markets, and supports a broad
range of commercial satellite telecommunications gateway products. These technologies and products
support two-way, high-speed space data communications for telemetry, satellite remote sensing, and
high-data-rate communications applications. The satellite antenna shown above is part of a system used
for high-speed data transmissions. (Courtesy of National Aeronautics and Space Administration.)
T
The Oscillator or Discrete Radiator
The basic building block for antenna analysis is a linear conductor. Movement of electrons (current) in the
conductor induces an electromagnetic field. When the electron motion is oscillatory—e.g., a dipole with
periodic electron motion, the induced electric field, E, is proportional to cos(wt – kx + f), where w is radian
frequency of oscillation, t is time, k is wave number, x is distance from the oscillator, and f is the phase associated
with this oscillator (relative to the time and spatial coordinate origins). When the analysis is restricted to a
fixed position x, the electric field can be expressed as
(38.25)
where the phase term f now includes the kx term, and all of the constants of proportionality are included in
the amplitude A. Basically, the assumption is that oscillating currents produce oscillating fields. The description
of a receiving antenna is analogous: an oscillating field induces a periodic current in the conductor.
The field from a pair of oscillators separated in phase by d radians is
(38.26)
Using phasor notation,
?
E
d
, the cosines are converted to complex exponentials and the radial frequency term,
wt, is suppressed,
(38.27)
The amplitude of the sinusoidal modulation E
d
(t) can be calculated as ê
?
E
d
?. The intensity is
(38.28)
When the oscillators are of the same amplitude, A = A
1
= A
2
, then
(38.29)
For a series of n equal amplitude oscillators with equal phase spacing
(38.30)
FIGURE 38.7 Examples of several types of antennas: (a) pyramidal horn, (b) conical horn, (c) axial slot on a cylinder,
and (d) parabolic reflector.
Et A t( ) cos( )=+wf
Et A t A t
d
wf wfd( ) cos ( ) cos ( )=+++
12
?
()
()
Et Ae Ae
ii
d
ffd
=+
+
12
IE A A AA==++** ** **
?
cos( )
d
d
2
1
2
2
2
12
2
EtAt At
At
d
wf wfd
d
wf
d
( ) cos( ) cos( )
cos cos
=++++
=
?
è
?
?
?
÷
++
?
è
?
?
?
÷
2
22
Et A t j
n
j
n
d
wf d( ) cos( )=++
=
-
?
0
1
? 2000 by CRC Press LLC
By using phasor arithmetic the intensity is given as
(38.31)
where I
0
=n
–2
to normalize the intensity pattern at d = 0.
For an incoming plane wave which is tilted at an angle q from the normal, the relative phase difference
between two oscillators is kd sinq, where d is the distance between oscillators and k is the wave number 2p/l
(see Fig. 38.8). For three evenly spaced oscillators, the phase difference between the end oscillators is 2kd sinq.
In general, the end-to-end phase difference for n evenly spaced oscillators is (n – 1)kd sinq. This formulation
is identical to the phase representation in Eq. (38.30) with d = kd sinq. Therefore, the intensity as a function
of incidence angle q for an evenly spaced array of n elements is
(38.32)
where L = nd corresponds to the physical dimension (length) of the aperture of oscillators. The zeros of this
function occur at kL sinq = 2mp, for any nonzero integer m. Equivalently, the zeros occur when sinq = ml/L.
When the element spacing d is less than a wavelength, the number of zeros for 0 < q < p/2 is given by the
largest integer M such that M ￡ L/l. Therefore, the ratio of wavelength to largest dimension, l/L, determines
both the location (in q space) and the number of zeros in the intensity pattern when d ￡ l. The number of
oscillators controls the amplitude of the side lobes.
For n = 1, the intensity is constant—i.e., independent of angle. For l > L, both the numerator and
denominator of Eq. (38.32) have no zeros and as the length of an array shortens (relative to a wavelength), the
intensity pattern converges to a constant (n = 1 case). As shown in Fig. 38.9, a separation of l/4 has an intensity
rolloff less than 1 dB over p/2 radians (a l/2 separation rolls off 3 dB). This implies that placing antenna
elements closer than l/4 does not significantly change the intensity pattern. Many microwave antennas exploit
this and use a mesh or parallel wire (for polarization sensitivity) design rather than covering the entire aperture
with conductor. This reduces both weight and sensitivity to wind loading. Note that the analysis has not
accounted for phase variations from position errors in the element placement where the required accuracy is
typically better than l/10.
FIGURE 38.8A two-element and an n-element array with equal spacing between elements. The propagation length
difference between elements is d sinq, which corresponds to a phase difference of kd sinq, where k is the wave number 2p/l.
The length L corresponds to a continuous aperture of length nd with the sample positions beginning d/2 from the ends.
ItE Ae e A
e
e
I
n
I
n
nn
iij
j
n in
i
dd
fd
d
d
d
d
d
d
()
?
cos( )
cos()
sin( )
sin( )
== =
-
-
=
-
-
=
=
-
?
**
2
0
1
2
2
2
0
0
2
2
1
1
1
1
2
2
/
/
II
knd
kd
I
kL
n
kL
I
L
L
n
nL
()
sin sin
sin sin
sin sin
sin sin
sin sin
sin sin
q
q
q
q
q
p
l
q
p
l
q
=
?
è
?
?
?
÷
?
è
?
?
?
÷
=
?
è
?
?
?
÷
?
è
?
?
?
÷
=
?
è
?
?
?
÷
?
è
?
?
?
÷
0
2
2
0
2
2
0
2
2
1
2
1
2
1
2
1
2
? 2000 by CRC Press LLC
For L > > l, sinq ? q, which implies that the first zero is at q = l/L. The location of the first zero is known
as the Rayleigh resolution criteria. That is, two plane waves separated by at least l/L radians can be discriminated.
For imaging applications, this corresponds roughly to the smallest detectable feature size. As shown in Fig. 38.10,
the first zero occurs at approximately l/L = 0.25 radians (the Rayleigh resolution). Note that there is no side
lobe suppression until d ￡ l, when the location of the zeros becomes fixed. Having more than eight array
elements (separation of less than a quarter wavelength) only moderately reduces the height of the maximum
side lobe.
Synthetic Apertures
In applications such as air- and space-based radar, size and weight constraints prohibit the use of very large
antennas. For instance, if the L-band (23.5-cm wavelength) radar imaging system on the Seasat satellite (800-km
altitude, launched in 1978) had a minimum resolution specification of 23.5 m, then, using the Rayleigh
resolution criteria, the aperture would need to be 8 km long. In order to attain the desired resolution, an
aperture is “synthesized” from data collected with a physically small (10 m) antenna traversing an 8-km flight
FIGURE 38.9Normalized intensity pattern in decibels (10 log(I)) for a two-element antenna with spacing 4l, 2l, l, l/2,
and l/4 between the elements.
FIGURE 38.10Normalized intensity pattern in decibels (10 log(I)) for a length 4l array with 2, 3, 4, 5, and 8 elements.
? 2000 by CRC Press LLC
path. Basically, by using a stable oscillator on the spacecraft, both amplitude and phase are recorded, which
allows postprocessing algorithms to combine the individual echoes in a manner analogous to an antenna array.
From an antenna perspective, an individual scattering element produces a different round trip propagation
path based on the position of the physical antenna—a synthetic antenna array. Using the geometry described
in Fig. 38.11, the phase is
(38.33)
It is convenient to assume a straight-line flight path along the x-axis, a planar earth (x, y plane), and a
constant velocity, v, with range and cross-range components v
r
(x) and v
c
(x), respectively. In many radar
applications the broad side distance to the center of the footprint, R, is much larger than the size of the footprint.
This allows the distance R(x) to be expanded about R resulting in
(38.34)
The first term in Eq. (38.34) is a constant phase offset corresponding to the center of beam range bin and
can be ignored from a resolution viewpoint. The second term, 2v
r
/l, is the Doppler frequency shift due to the
relative (radial) velocity between antenna and scattering element. The third term represents a quadratic cor-
rection of the linear flight path to approximate the constant range sphere from a scattering element. It is worth
noting that synthetic aperture systems do not require the assumptions used here, but accurate position and
motion compensation is required.
For an antenna with cross range dimension D and a scattering element at range R, the largest synthetic
aperture that can be formed is of dimension lR/D (the width of the footprint). Because this data collection
scenario is for round trip propagation, the phase shift at each collecting location is twice the shift at the edges
of a single physical antenna. Therefore at a range R, the synthetic aperture resolution is
(38.35)
The standard radar interpretation for synthetic apertures is that information coded in the Doppler frequency
shift can be decoded to produce high-resolution images. It is worth noting that the synthetic aperture can be
formed even with no motion (zero Doppler shift). For the no-motion case the antenna array interpretation is
appropriate. This approach has been used for acoustic signal processing in nondestructive evaluation systems as
FIGURE 38.11Synthetic aperture radar geometry and nearly orthogonal partitioning of the footprint by range (circular)
and Doppler frequency (hyperbolic) contours.
f
p
l
p
l
() ()xRxxyz==++
2
2
2
2
222
f
p
l
p
lll
() ()tRvt
Rv
t
v
R
t
rc
==++
ì
í
?
?
?
ü
y
?
t
?
2
22
22
2
2
ll
l
R
D
R
RD
D
SA
==
22/
? 2000 by CRC Press LLC
well as wave migration codes for seismic signal processing. When there is motion, the Doppler term in the
expansion of the range dominates the phase shift and therefore becomes the useful metric for predicting resolution.
Geometric Designs
The phase difference in a linear array was caused by the spatial separation and allowed the discrimination of
plane waves arriving at different angles. Desired phase patterns can be determined by using analytic geometry
to position the elements. For example, if coherent superposition across a wave front is desired, the wave front
can be directed (reflected, refracted, or diffracted) to the receiver in phase. For a planar wave front, this
corresponds to a constant path length from any point on the reference plane to the receiver. Using the geometry
in Fig. 38.12, the sum of the two lengths (x, R + h) to (x, y) and (x, y) to (0, h) must be a constant independent
of x—which is R + 2h for this geometry. This constraint on the length is
(38.36)
This is the equation for a parabola. Losses would be minimized if the wave front were specularly reflected to
the transceiver. Specular reflection occurs when the angles between the normal vector N [or equivalently the
tangent vector T = (x,f￠￠(x))] = (1,x/2h) and the vectors A = (0, –1) and B = (–x, h – y) are equal. This is the
same as equality of the inner products of the normalized vectors, which is shown by
The constant path length and high gain make the parabolic antenna popular at many wavelengths including
microwave and visible. More than one reflecting surface is allowed in the design. The surfaces are typically
conical sections and may be designed to reduce a particular distortion or to provide better functionality.
Compound designs often allow the active elements to be more accessible and eliminate long transmission lines.
A two-bounce reflector with a parabolic primary and a hyperbolic secondary is known as a Cassegrain system.
In all reflector systems it is important to account for the blockage (“shadow” of the feed, secondary reflector,
and support structures) as well as the spillover (radiation propagating past the intended reflecting surface).
Continuous Current Distributions (Fourier Transform)
Ideally, antennas would be designed using solutions to Maxwell’s equations. Unfortunately, in most cases exact
analytic and numerical solutions to Maxwell’s equations are difficult to obtain. Under certain conditions,
FIGURE 38.12Parabolic reflector systems: (a) geometry for determining the function with a constant path length and
specular reflection, (b) single-bounce parabolic reflector, (c) two-bounce reflector with a parabolic primary and hyperbolic
secondary (Cassegrain).
Rhyxhy Rh x hy+-+ +-=+ =
22
24() or
2
??
(,)
(,)
??
(,) (, )
()
()
()
TA
TB
×=
+
×-=
-
+
×=
+
×
--
+-
=
-+
+
=
-
+
2
4
01
4
2
4
4
4
4
22 22
222 2
22
2232
22
hx
xh
x
xh
hx
xh
xhy
xhy
xx h
xh
x
xh
/
(38.37)
(38.38)
? 2000 by CRC Press LLC
approximations can be introduced that allow solution to the wave equations. Approximating spherical wave
fronts as quadratics has been shown for the synthetic aperture application and is valid when the propagation
distance is greater than (pL
2
/4l)
1/3
, where L is the aperture size. In general, this is known as the Fresnel or
near-field approximation. When the propagation distance is at least 2L
2
/l, the angular radiation pattern can
be approximated as independent of distance from the aperture. This pattern is known as the normalized far-
field or Fraunhofer distribution, E(q), and is related to the normalized current distributed across an antenna
aperture, i(x), by a Fourier transform:
(38.39)
where u = sinq and x￠ = x/l.
Applying the Fraunhofer approximation to a line source of length L
(38.40)
which is Eq. (38.32) when n >> L/l. As with discrete arrays, the ratio L/l is the important design parameter:
sinq = l/L is the first zero (no zeros for l > L) and the number of zeros is the largest integer M such that M ￡ L/l.
In two dimensions, a rectangular aperture with uniform current distribution produces
(38.41)
The field and intensity given in Eq. (38.41) are normalized. In practice, the field is proportional to the aperture
area and inversely proportional to the wavelength and propagation distance.
The normalized far-field intensity distribution for a uniform current on a circular aperture is a circularly
symmetric function given by
(38.42)
where J
1
is the Bessel function of the first kind, order one. This far-field intensity is called the Airy pattern. As
with the rectangular aperture, the far-field intensity is proportional to the square of the area and inversely
proportional to the square of the wavelength and the propagation distance. The first zero (Rayleigh resolution
criteria) of the Airy pattern occurs for uL/l = 1.22 or sinq = 1.22l/L. As with linear and rectangular apertures,
the resolution scales with l/L.
Figure 38.13 shows a slice through the normalized far-field intensity of both a rectangular aperture and a
circular aperture. The linearity of the Fourier transform allows apertures to be represented as the superposition
of subapertures. The primary reflector, the obscurations from the support structures, and the secondary reflector
Eu ixe dx
iux
() ()= ￠￠
￠
ò
2p
Eu e dx
L
u
L
u
L
L
L
iux
L
L
( sin)
sin sin sin
sin
== ￠=
?
è
?
?
?
÷
=
?
è
?
?
?
÷
￠
-
ò
q
p
l
p
l
p
l
q
p
l
q
p
l
l
2
2
2
/
/
Euu
uL
uL
uL
uL
Iuu Eu Eu
RRL
(,)
sin sin
(,) () ()
12
11
11
22
22
12 1
2
2
2
=
?
è
?
?
?
÷
?
è
?
?
?
÷
=
p
l
p
l
p
l
p
l
and ** *
Iu
JuL
uL
C
()=
?
è
?
?
?
÷
é
?
ê
ê
ê
ê
ê
ù
?
ú
ú
ú
ú
ú
2
1
2
p
l
p
l
? 2000 by CRC Press LLC
of a Cassegrain-type antenna can be modeled. Numerical evaluation of the Fourier transform permits straight-
forward calculation of the intensity patterns, even for nonuniform current distributions.
Antenna Parameters
Direct solutions to Maxwell’s equations or solutions dependent on approximations provide the analytic tools
for designing antennas. Ultimately, the analysis must be confirmed with experiment. Increasingly sensitive radar
and other antenna applications have resulted in much more attention to edge effects (from the primary aperture,
secondary, and/or support structures). The geometric theory of diffraction as well as direct Maxwell solvers
are making important contributions.
With the diversity of possible antenna designs, a collection of design rules of thumb are useful. The directivity
and gain for a few popular antenna designs are given in Table 38.1. Directivity is the ratio of the maximum to
average radiation intensity. The gain is defined as the ratio of the maximum radiation intensity from the subject
antenna to the maximum radiation intensity from a reference antenna with the same power input. The
directivity, D, and gain, G, of an antenna can be expressed as
(38.43)
FIGURE 38.13Normalized intensity pattern in decibels (10 log[I(v = uL/l)]) for a rectangular and a circular antenna
aperture with uniform current distributions.
TABLE 38.1Directivity and Gain of Some Higher Frequency Antennas
Antenna Type Directivity
a
Gain
a
Uniform rectangular aperture
Large square aperture
Large circular aperture (parabolic reflector)
Pyramidal horn
a
Directivity and gain are relative to a half-wave dipole.
4
2
p
l
LL
xy
4
2
p
l
LL
xy
126
2
.
L
l
?
è
?
?
?
÷
77
2
.
L
l
?
è
?
?
?
÷
987
2
.
D
l
?
è
?
?
?
÷
7
2
D
l
?
è
?
?
?
÷
4
2
p
l
?
è
?
?
?
÷
LL
xy
05
4
2
.
p
l
?
è
?
?
?
÷
LL
xy
DA GA
em e
=
?
è
?
?
?
÷
?
è
?
?
?
÷
44
22
p
l
p
l
and =
? 2000 by CRC Press LLC
where A
em
is the maximum effective aperture and A
e
is the actual effective aperture of the antenna. Because of
losses in the system, A
e
= kA
em
, where k is the radiation efficiency factor. The gain equals the directivity when
there are no losses (k = 1), but is less than the directivity if there are any losses in the antenna (k < 1), that is,
G = kD.
As an example, consider the parabolic reflector antenna where efficiency degradation includes
? Ohmic losses are small (k = 1)
? Aperture taper efficiency (k = 0.975)
? Spillover (feed) efficiency (k = 0.8)
? Phase errors in aperture field (k = 0.996 to 1)
? Antenna blockage efficiency (k = 0.99)
? Spar blockage efficiency (k = 0.994)
Each antenna system requires a customized analysis of the system losses in order to accurately model performance.
Defining Terms
Antenna: A physical device for transmitting or receiving propagating waves.
Aperture antenna: An antenna with a physical opening, hole, or slit. Contrast with a wire antenna.
Array antenna: An antenna system performing as a single aperture but composed of antenna subsystems.
Directivity: The ratio of the maximum to average radiation intensity.
Fraunhofer or far field: The propagation region where the normalized angular radiation pattern is indepen-
dent of distance from the source. This typically occurs when the distance from the source is at least
2L
2
/l, where L is the largest dimension of the antenna.
Fresnel or near field: The propagation region where the normalized radiation pattern can be calculated using
quadratic approximations to the spherical Huygens’ wavelet surfaces. The pattern can depend on distance
from the source and is usually valid for distances greater than (p/4l)
1/3
L
2/3
, where L is the largest
dimension of the antenna.
Gain: The ratio of the maximum radiation intensity from the subject antenna to the maximum radiation
intensity from a reference antenna with the same power input. Typical references are a lossless isotropic
source and a lossless half-wave dipole.
Oscillator: A physical device that uses the periodic motion within the material to create propagating waves.
In electromagnetics, an oscillator can be a conductor with a periodic current distribution.
Reactive near field: The region close to an antenna where the reactive components of the electromagnetic
fields from charges on the antenna structure are very large compared to the radiating fields. Considered
negligible at distances greater than a wavelength from the source (decay as the square or cube of distance).
Reactive field is important at antenna edges and for electrically small antennas.
Related Topic
37.1 Space Propagation
References
R. Feynman, R.B. Leighton, and M.L. Sands, The Feynman Lectures on Physics, Reading, Mass.: Addison-Wesley,
1989.
J.P. Fitch, Synthetic Aperture Radar, New York: Springer-Verlag, 1988.
J.W. Goodman, Introduction to Fourier Optics, New York: McGraw-Hill, 1968.
H. Jasik, Antenna Engineering Handbook, New York: McGraw-Hill, 1961.
R.W.P. King and G.S. Smith, Antennas in Matter, Cambridge: MIT Press, 1981.
J.D. Krause, Antennas, New York: McGraw-Hill, 1950.
Y.T. Lo and S.W. Lee, Antenna Handbook, New York: Van Nostrand Reinhold, 1988.
? 2000 by CRC Press LLC
A.W. Rudge, K. Milne, A.D. Olver, and P. Knight, The Handbook of Antenna Design, London: Peter Peregrinus,
1982.
M. Skolnik, Radar Handbook, New York: McGraw-Hill, 1990.
B.D. Steinberg, Principles of Aperture & Array System Design, New York: John Wiley & Sons, 1976.
Further Information
The monthly IEEE Transactions on Antennas and Propagation as well as the proceedings of the annual IEEE
Antennas and Propagation International Symposium provide information about recent developments in this
field. Other publications of interest include the IEEE Transactions on Microwave Theory and Techniques and the
IEEE Transactions on Aerospace and Electronic Systems.
Readers may also be interested in the “IEEE Standard Test Procedures for Antennas,” The Institute for
Electrical and Electronics Engineers, Inc., ANSI IEEE Std. 149-1979, 1979.
38.3Microstrip Antennas
David M. Pozar
Introduction
Microstrip antenna technology has been the most rapidly developing topic in the antenna field in the last 15
years, receiving the creative attentions of academic, industrial, and government engineers and researchers
throughout the world. As a result, microstrip antennas have quickly evolved from a research novelty to com-
mercial reality, with applications in a wide variety of microwave systems. Rapidly developing markets in personal
communications systems (PCS), mobile satellite communications, direct broadcast television (DBS), wireless
local-area networks (WLANs), and intelligent vehicle highway systems (IVHS) suggest that the demand for
microstrip antennas and arrays will increase even further.
Although microstrip antennas have proved to be a significant advance in the established field of antenna
technology, it is interesting to note that it is usually their nonelectrical characteristics that make microstrip
antennas preferred over other types of radiators. Microstrip antennas have a low profile and are light in weight,
they can be made conformal, and they are well suited to integration with microwave integrated circuits (MICs).
If the expense of materials and fabrication is not prohibitive, they can also be low in cost. When compared
with traditional antenna elements such as wire or aperture antennas, however, the electrical performance of
the basic microstrip antenna or array suffers from a number of serious drawbacks, including very narrow
bandwidth, high feed network losses, poor cross polarization, and low power-handling capacity. Intensive
research and development has demonstrated that most of these drawbacks can be avoided, or at least alleviated
to some extent, with innovative variations and extensions to the basic microstrip element [James and Hall,
1989; Pozar and Schaubert, 1995]. Some of the basic features of microstrip antennas are listed below:
?Low profile form factor
?Potential light weight
?Potential low cost
?Potential conformability with mounting structure
?Easy integration with planar circuitry
?Capability for linear, dual, and circular polarizations
?Versatile feed geometries
Basic Microstrip Antenna Element
The basic microstrip antenna element is derived from a l
g
/2 microstrip transmission line resonator [Pozar,
1990]. It consists of a thin metallic conducting patch etched on a grounded dielectric substrate, as shown in
Fig. 38.14. This example is shown with a coaxial probe feed, but other feeds are possible, as discussed below.
? 2000 by CRC Press LLC
The patch has a length L along the x-axis, and width W along the y-axis. The dielectric substrate has a thickness
d and a dielectric constant e
r
, and is backed with a conducting ground plane. With a coaxial probe feed, the
outer conductor of the coaxial line is connected to the ground plane, and the inner conductor is attached to
the patch element. The position of the feed point relative to the edge of the patch controls the input impedance
level of the antenna. In operation, the length of the patch element is approximately l
g
/2, forming an open-
circuit resonator. Because the patch is relatively wide, the patch edges at x = –L/2 and L/2 effectively form slot
apertures which radiate in phase to form a broadside radiation pattern.
Many analytical models have been developed for the impedance and radiation properties of microstrip
antennas [James and Hall, 1989], but most of the qualitative behavior of the element can be demonstrated
using the relatively simple transmission line model. As shown in Fig. 38.15, the patch element is modeled as a
length, L, of microstrip transmission line of characteristic impedance Z
0
. The characteristic impedance of the
line can be found using simple approximations [Pozar, 1990] and is a function of the width, W, of the line as
well as the substrate thickness and dielectric constant. The ends of the transmission line are terminated in
FIGURE 38.14 Geometry of rectangular coaxial probe-fed microstrip antenna.
FIGURE 38.15 Transmission line circuit model for a rectangular microstrip antenna. The feed point is positioned a
distance s from the radiating edge of the patch.
? 2000 by CRC Press LLC
admittances, Y = G + jB, where the conductance serves to model the radiation from the ends of the line, and
the susceptance serves to model the effective length extension of the line (due to fringing fields). Several
approximations are available for calculating the end admittances [James and Hall, 1989], with a typical result
for d << l
0
given as
(38.44)
where k
0
= 2p/l
0
and h = . The susceptance B is typically positive, implying a capacitive end effect. This
means that the resonant length of the patch will be slightly less than l
g
/2. If the feed probe is located a distance
s from the edge of the patch, the input impedance seen by the probe can be calculated using basic transmission
line theory from the circuit of Fig. 38.15. Resonance is defined as the frequency at which the imaginary part
of the input impedance is zero.
As a result of the symmetry of the transmission line resonator, the voltage along the transmission line will
have maxima at the ends and a null at the center of the line. This implies that the input impedance will be
maximum when the feed point is at the edge of the patch, and will decrease to zero as the feed is moved to the
center of the patch. Fig. 38.16 shows a Smith chart plot of the input impedance of a coaxial probe-fed microstrip
antenna vs. frequency, for three different probe positions. Observe that the input impedance locus decreases
as the feed point moves toward the center of the patch. Also, observe that the impedance locus becomes more
inductive as the feed point moves toward the center of the patch.
The far-field radiation patterns can also be derived from the transmission line model by treating the radiating
edges at x = –L/2 and L/2 as equivalent slots. In the coordinate system of Fig. 38.14, the normalized far-zone
fields of the rectangular patch can be expressed as
FIGURE 38.16 Smith chart plot of the input impedance of a probe-fed rectangular microstrip antenna vs. frequency, for
three different feed positions. Patch parameters are L = 2.5 cm, W = 3.0 cm, e
r
= 2.2, d = 0.79 cm. Frequency sweep runs
from 3.6 to 4.25 GHz, in steps of 50 MHz.
YGjB
kW
jkd=+= +-
( )
[]
0
0
0
2
11064
h
.ln
m
0
0
e
? 2000 by CRC Press LLC
(38.45a)
(38.45b)
where
and q and f are spherical coordinates. These patterns have maxima broadside (q = 0) to the patch, with 3-dB
beamwidths typically in the range of 90° to 120°. Typical E-plane (f = 0) and H-plane (f = 90°) microstrip
antenna radiation patterns are shown in Fig. 38.17.
Microstrip antenna elements have a number of useful and inter-
esting features, but probably the most serious limitation of this
technology is the narrow bandwidth of the basic element. While
antenna elements such as dipoles, slots, and waveguide horns have
operating bandwidths ranging from 15 to 50%, the traditional
microstrip patch element typically has an impedance bandwidth of
only a few percent. Fig. 38.18 shows the impedance bandwidth vs.
substrate thickness for a rectangular microstrip antenna with sub-
strate permittivities of 2.2 and 10.2. Observe from the figure that
bandwidth decreases as the substrate becomes thinner and as the
dielectric constant increases. Both of these trends are explained as
a result of the increased Q of the resonator, basically due to the fact
that the patch current is in close proximity to its negative image in
the substrate ground plane. In terms of bandwidth, it is preferable
to use a thick antenna substrate, with a low dielectric constant. But
because of inductive loading and possible spurious radiation from
coplanar microstrip circuitry, the thickness of a microstrip antenna substrate is typically limited to 0.02l or
less. This illustrates the essential compromise associated with the microstrip antenna concept, as it is not possible
to obtain optimum performance from both a microstrip antenna and microstrip circuitry on a single dielectric
substrate. These two functions are distinct electromagnetically, since the bound fields associated with nonra-
diating circuitry obviate efficient radiation.
While the bandwidth of the basic element is limited, considerable research and development during the last
15 years has led to a number of creative and novel techniques for the enhancement of microstrip antenna
bandwidth, so that impedance bandwidths ranging from 10 to 40% can now be achieved [James and Hall,
1989; Pozar and Schaubert, 1995]. While there have been dozens of proposed techniques for the enhancement
of microstrip antenna bandwidth, they can all be categorized according to three canonical approaches:
?Impedance matching using matching network
?Introducing dual resonance with stacked or parasitic elements
?Reducing efficiency by adding lossy elements
The reader is referred to the literature for more details on specific techniques for bandwidth improvement.
Figure 38.18 also shows the efficiency of the antenna, defined as
EE
q
a
a
bf=
0
sin
cos cos
EE
f
a
b
bqf=
0
sin
cos cos sin
aqf
bqf
=
=
kW
kL
0
0
2
2
sin sin
sin cos
FIGURE 38.17 E- and H-plane far-field
radiation patterns of the rectangular micros-
trip antenna of Fig. 38.16.
? 2000 by CRC Press LLC
(38.46)
where P
rad
is the radiated power and P
loss
is the power lost in the antenna. Losses in a microstrip antenna occur
in three ways: conductor loss, dielectric loss, and surface wave loss. Unless the substrate is extremely thin,
conductor loss is generally negligible. For quality microwave substrates (loss tangent ￡ 0.002), dielectric loss is
also relatively small. Surface waves, which are fields bound to the dielectric substrate that propagate along its
surface, often account for the dominant loss mechanism for microstrip antennas. As can be seen in Fig. 38.18,
efficiency decreases with increasing substrate thickness and dielectric constant, again suggesting the use of low-
dielectric-constant substrates. The overall radiation efficiency of a microstrip antenna on a low-dielectric
substrate is typically 95%, or better.
Besides rectangular patch elements, it is possible to use a variety of other patch shapes as resonant radiating
elements. For purposes of polarization purity and analytical simplicity, however, it is usually preferable to use
rectangular, square, or circular elements. Linear polarization is best obtained with rectangular elements, while
dual linear or circular polarization can be obtained with square or circular patch elements [James and Hall,
1989; Pozar and Schaubert, 1995].
Feeding Techniques for Microstrip Antennas
While Fig. 38.14 shows a coaxial probe-fed microstrip antenna element, it is also possible to feed the patch
element by several other methods. Fig. 38.19a shows a rectangular patch element fed with a microstrip trans-
mission line coplanar with the patch element. The amount of inset of the feed line controls the input impedance
level at resonance, in a manner analogous to the positioning of the coax probe feed for impedance control.
The equivalent circuit of the antenna near resonance is also shown in the figure. The patch appears as a parallel
RLC resonant circuit, with a series inductance that represents the near-field effect of the microstrip feed line.
(The same equivalent circuit applies to the probe-fed microstrip antenna.) Both the probe feed and the line
feed excite the patch element through coupling between the equivalent J
z
electric current of the feed and the
E
z
directed field of the patch resonator, which has a maximum below the center of the patch.
The direct-contacting coax probe and inset microstrip line feeds have the advantage of simplicity, but suffer
from some disadvantages. First, bandwidth is limited because of the requirement of a thin substrate, as discussed
above. In addition, the inherent E-plane asymmetry of these feeds generates higher-order modes which lead
FIGURE 38.18 Impedance bandwidth and radiation efficiency of a microstrip antenna vs. substrate thickness, for two
values of substrate permittivity.
e
P
PP
=
+
rad
rad loss
? 2000 by CRC Press LLC
to cross polarization. And, in the case of the coax feed, the need for soldering can decrease reliability and
increase cost if a large number of elements are involved.
It is also possible to feed a microstrip antenna element using noncontacting feeds of various forms. Fig. 38.19b
shows a proximity feed, where a two-layer substrate houses an embedded microstrip transmission feed line,
with the radiating patch located on the top of a substrate layer placed over the microstrip feed line. The feed
line is terminated in an open-circuited stub below the patch. Proximity coupling (often referred to in the
literature by the less-descriptive term electromagnetic coupling) has the advantage of allowing the patch to reside
on a relatively thick substrate, for enhanced bandwidth, while the feed line sees an effectively thinner substrate,
FIGURE 38.19 Three types of feeding methods for rectangular microstrip antennas, and their associated equivalent circuits:
(a) patch fed with an inset microstrip transmission line, (b) patch fed by proximity coupling to a microstrip transmission
line, (c) patch fed by aperture coupling to a microstrip transmission line.
? 2000 by CRC Press LLC
which is preferred to minimize spurious radiation and coupling. Fabrication is a bit more difficult than the
single-layer coax or line feed, because of the requirement of bonding and aligning two substrates. The equivalent
circuit of the proximity-coupled element is shown in Fig. 38.19b, where the series capacitor is indicative of the
capacitive nature of the coupling between the open-ended microstrip line and the patch element.
Another type of noncontacting feed is the aperture-coupled microstrip antenna shown in Fig. 38.19c [James
and Hall, 1989; Pozar and Schaubert, 1995]. This configuration consists of two parallel substrates separated by
a ground plane. A microstrip feed line on the bottom of the bottom substrate is coupled through a small
aperture (typically a thin slot) in the ground plane to a microstrip patch element on the top of the top substrate.
This arrangement allows a thin, high-dielectric-constant substrate to be used for the feed line, and a thicker,
low-dielectric-constant substrate to be used for the radiating element. In this way, the two-layer design of the
aperture-coupled element allows the substrates to be optimized for the distinct functions of circuit components
and radiating elements. In addition, the ground plane provides isolation between the radiating aperture and
possible spurious radiation or coupling from the feed network. An important aspect of the aperture-coupled
approach is that the coupling aperture is below resonant size, so that the backlobe radiated by the slot is typically
15 to 20 dB below the main forward beam.
The aperture-coupled geometry affords several degrees of freedom for control of the electrical properties of
the antenna. The slot size (length) primarily determines the coupling level and, hence, the input impedance.
Tightest coupling occurs when the slot is centered below the patch, with the input impedance decreasing as
the slot size is decreased. As with any microstrip antenna, the resonant frequency is controlled primarily by
the length of the patch. The feed line stub length can be used to adjust the reactance loading of the element,
and the antenna substrate thickness and dielectric constant has a direct effect on the bandwidth of the element.
It is also possible to use the slot to provide a double tuning effect to increase significantly impedance bandwidth.
Such aperture-coupled antennas have been demonstrated with bandwidths up to 40% [Pozar and Schaubert,
1995]. Another unique feature of the aperture coupled patch is that the principal plane patterns have theoret-
ically zero cross polarization because of the symmetry of the element.
Microstrip Antenna Arrays
One of the most useful features of microstrip antenna technology is the ease with which array antennas can
be constructed, since the feed network can be fabricated with microstrip transmission lines and microstrip
circuit components at the same time as the microstrip radiating elements. This eliminates the cumbersome
and expensive coaxial or waveguide feed networks that are necessary for other types of arrays. In fact, microstrip
technology offers such versatility in design that a wide variety of series-fed, corporate-fed, fixed-beam, scanning,
multilayer, and polarization agile microstrip arrays have been demonstrated, many examples of which can be
found in the literature [James and Hall, 1989; Pozar and Schaubert, 1995].
One of the most convenient architectures for microstrip arrays is the single-layer design where the microstrip
feed lines are printed on the same substrate layer as the radiating patch elements. This results in a simple, low-
profile, inexpensive, and easily fabricated antenna assembly. An example of a 2 ′ 4 microstrip array using this
type of configuration is shown in Fig. 38.20. The microstrip feed network consists of a main feed line driving
three levels of coplanar two-way power dividers, which in turn drive eight edge-fed patches. This is an example
of a corporate feed network, in contrast to a series type of feed where array elements are tapped off of a single
microstrip line. The corporate feed provides good bandwidth and allows precise control of element excitation,
but requires considerable substrate area and can be lossy. A series feed can be very compact and efficient, but
its bandwidth is typically limited to a few percent. Both types of feeds can be used for single and dual
polarizations. More flexibility for feed network layout can be obtained by using a two-sided aperture-coupled
patch geometry. This allows the feed network to be isolated from the radiating aperture by the ground plane,
and the extra substrate area can be very useful for arrays that require dual polarization or dual-frequency
operation. Similar features can be obtained by using feedthrough pins or vias, but the added fabricational
complexity of solder connections can be formidable in a large array.
A serious limitation of microstrip array technology is that array gain is limited by the relatively high losses
of microstrip transmission lines. While array directivity increases with the area of the radiating aperture, the
losses of the feed network increase exponentially with array size. This is especially serious at higher frequencies,
? 2000 by CRC Press LLC
where it is usually impractical to design a microstrip array with a coplanar feed network for gains in excess of
28 to 32 dB. Off-board feed networks or multilayer designs can be used to partially circumvent this problem,
but at the expense of simplicity and cost.
Computer-Aided Design for Microstrip Antennas
Ideally, antenna CAD software would combine a user-friendly interface with a computationally efficient set of
accurate and versatile theoretical models. While software with such features has reached a fairly high level of
refinement for the analysis of low-frequency and microwave circuit analysis and optimization, the development
of microstrip antenna CAD software lags far behind. One reason is the economic reality that the market for
antenna software is relatively small, which perhaps explains why there is very little commercially available
antenna CAD software of any type. Microstrip antenna CAD software development has also been slow because
of the fact that such antennas are relatively new, receiving serious attention only during the last 15 years.
Furthermore, microstrip antenna geometries are relatively difficult to model because of the presence of dielectric
inhomogeneities and a wide variety of feeding techniques and other geometric features. This last consideration
makes the development of a general-purpose microstrip antenna analysis package extremely difficult.
It may come as a surprise to the newcomer to practical antenna development, but it must be realized that
many microstrip antenna designs have been successfully completed with little or no CAD support. There are,
however, many situations where antennas and arrays can be designed more effectively, with better performance
and less experimental iteration, when proper CAD software tools are available. And there are situations involving
large arrays of microstrip elements which critically rely on the use of CAD software for successful design. Thus,
CAD software is not absolutely necessary for all facets of microstrip antenna design work, but good software
tools can be very useful for dealing with the more-complicated microstrip geometries. Another point that seems
to be especially true for antenna design in general is that CAD software, no matter how versatile or accurate,
cannot substitute for experience and understanding of the fundamentals of antenna operation. Further discus-
sion of microstrip antenna CAD issues can be found in [Pozar and Schaubert, 1995].
Defining Terms
Array antenna: A repetitive grouping of basic antenna elements functioning as a single antenna with improved
gain or pattern characteristics.
Bandwidth: The fractional frequency range over which the impedance match or pattern qualities of an
antenna meet a required specification.
Beamwidth: The angular width of the main beam of an antenna, typically measured at either –3 or –10 dB
below beam maximum.
Microstrip line: A planar transmission line consisting of a conducting strip printed (or etched) on a grounded
dielectric substrate.
Microwave integrated circuit (MIC): A microwave or RF subsystem formed by the monolithic or hybrid
integration of active devices, transmission lines, and related components.
FIGURE 38.20 A corporate-fed eight-element microstrip array antenna.
? 2000 by CRC Press LLC
Related Topics
37.1 Space Propagation ? 37.2 Waveguides
References
J. R. James and P. S. Hall, Eds., Handbook of Microstrip Antennas, London: Peter Peregrinus (IEE), 1989.
D. M. Pozar, Microwave Engineering, Reading, Mass: Addison-Wesley, 1990.
D. M. Pozar and D. H. Schaubert, Microstrip Antennas: The Analysis and Design of Microstrip Antennas and
Arrays, New York: IEEE Press, 1995.
Further Information
The most up-to-date information for developments in the field of microstrip antennas can be found in the
technical journals. These include the IEEE Transactions on Antennas and Propagation, the IEE Proceedings, Part
H, and Electronics Letters. There are also a large number of symposiums and conferences on antennas that
usually emphasize practical microstrip antenna technology, such as the IEEE International Symposium on
Antennas and Propagation, the International Symposium on Antennas and Propagation (ISAP–Japan), and the
International Conference on Antennas and Propagation (ICAP–Great Britain). Coverage of basic antenna theory
and design can be found in the previous sections in this chapter, as well as the references listed there.
? 2000 by CRC Press LLC
