Electric field lines
of radiated wave
Antenna
Transmission line
Guided EM wave Generator
Transition
region
Wave launched
into free space
(a) Transmission mode
Antenna
Transmission line
Receiver
Guided EM wave
Incident
wave
Transition
region
(b) Reception mode
Figure 3-1: Antenna as a transducer between a guided
electromagnetic wave and a free-space wave, for both
transmission and reception.
Circular
plate
reflector
(a) Thin dipole
(b) Biconical dipole
(c) Loop
(e) Periodic
(d) Helix
Radiating strip
Coaxial feed Dielectric
substrate
Phase shifters
Feed
point
Ground metal
plane
(f) Parabolic dish
reflector
(g) Horn
(h) Microstrip
Figure 3-2: Various types of antennas.
(i) Antenna array
Source
R
Transmitting
antenna
Receiving
antenna
Spherical wave
Plane-wave
approximation
Figure 3-3: Far-field plane-wave approximation.
z
Q = (R, θ, ϕ)
θ
R
i(t)
y
l
i(t)
x
ϕ
Figure 3-4: Short dipole placed at the origin of a
spherical coordinate system.
θ = 0°
z
Direction (θ, ϕ)
θ
θ = 90°
ϕ = 270°
Radiation
source
R
θ = 90° y
ϕ = 90°
ϕ
x
θ = 90°
ϕ = 0°
θ = 180°
Figure 3-5: Spherical coordinate system.
z
θ1 = 45°
F(θ)
θ
Dipole
1
θ = 90°
β = 90° 1
0
0.5
θ2 = 135°
(a) Elevation pattern
y
1
F(ϕ)
ϕ
1
0
x
Dipole
(b) Azimuth pattern
Figure 3-6: Radiation patterns of a short dipole.
z
R sin θ dϕ
R dθ
dA = R2 sin θ dθ dϕ
= R2 dΩ
Elevation
plane
θ R
y
ϕ
R dϕ
Azimuth plane
x
Figure 3-7: Definition of solid angle dΩ = sin θ d θ d φ .
Normalized radiation intensity (dB)
0
−5
−10
−15
−20
−25
−30
88
th
ni
Ze
89
2
1
θ
le
g
an
90
0
91
s)
ee
r
eg
(d
eϕ
ngl
a
uth
zim
−1
92 −2
A
es)
gre
e
d
(
Figure 3-8: Three-dimensional pattern of a narrowbeam antenna.
−10
30
Zenith angle θ (degrees)
−20
40
−30
50
60
10
20
0
−3
−5
Mainlobe
30
First sidelobe
40
Minor
lobes
−40
50
60
70
70
80
80
90
90
100
100
110
110
Backlobes
Normalized radiation intensity F(θ), (dB)
0
Normalized radiation intensity (dB)
10
20
−10
β1/2
θ = 0°
z
ϕ = 180°
θ
x
θ = 90°
ϕ = 0°
θ = 180°
−15
−20
−25
βnull
−30
120
120
130
130
140 150 160 170 180 170 160 150 140
(a) Polar diagram
−35
−50 −40 −30 −20 −10 θ1 0 θ2 10 20 30
Zenith angle θ (degrees)
40
50
(b) Rectangular plot
Figure 3-9: Representative plots of the normalized radiation pattern of a microwave antenna in (a) polar form and
(b) rectangular form.
1
F(θ, ϕ)
1
F = 1 within
the cone
Ωp
(a) Actual pattern
(b) Equivalent solid angle
Figure 3-10: The pattern solid angle Ωp defines an
equivalent cone over which all the radiation of the actual
antenna is concentrated with uniform intensity equal to
the maximum of the actual pattern.
z
0 dB
βxz
βyz
y
x
Figure 3-11: The solid angle of a unidirectional
radiation pattern is approximately equal to the product of
the half-power beamwidths in the two principal planes;
that is, Ωp ≈ βxz βyz .
Incident
wave
i
Pint
ZL
Load
Antenna
Figure 3-12: Receiving antenna represented by an
equivalent circuit.
At
Ar
R
Pt
Prad
Transmitting
antenna
Prec
Pint
Receiving
antenna
Figure 3-13: Transmitter–receiver configuration.
Pt
Prec
R1
θ
θ
R2
(a) Reflection by specular surface
Pt
R1
θ
θ
R2
Prec
(b) Image method
Figure 3-14: Reflection by a specular surface is
equivalent to direct transmission along a path of length
(R1 + R2 ), but modified by the reflectivity of the surface
Γ p (θ ).
xa
Ea(xa, ya)
R
θ
ya
Q
z
Observation
sphere
(a) A horn antenna with aperture field
distribution Ea(xa, ya)
d
Imaginary aperture
(b) Radiation by a parabolic dish reflector
illuminated by a small horn antenna
Figure 3-15: (a) A horn antenna with aperture field
distribution Ea (xa , ya ), and (b) radiation by a parabolic
dish reflector illuminated by a small horn antenna.
xa
Aperture
illumination
ly
lx
dxa
dya
s
R
z
x
Q
θ
ϕ
ya
y
Aperture plane A
Observation plane O
Figure 3-16: Radiation by an aperture in the xa –ya plane at z = 0.
F(γ)
0 dB
z
θ
R
Q = (R, θ)
ya
ly
xa
–3
–5
lx
–10
βxz
–13.2 dB
–15
–20
–25
–3
–2
–1
–30
1
γ = (lx/λ) sin θ
2
3
Figure 3-17: Normalized radiation pattern of a
uniformly illuminated rectangular aperture in the x–z
plane (φ = 0).
Sidelobes
d
β≈ λ
d
Boresight
(a) Pencil beam
βxz ≈ λ
lx
βyz ≈ λ
ly
ly
lx
(b) Fan beam
Figure 3-18: Radiation patterns of (a) a circular
reflector and (b) a cylindrical reflector (sidelobes not
shown).
xa
a
ra
ϕa
R
θ
a
z
ya
(a) Circular aperture in the xa-ya plane
F[dB]
0
−3
−5
−10
−15
−20
−25
−10 −8 −6 −4 −2 0
2
v = (2πa/λ) sin θ
4
6
8
10
(b) Radiation pattern
Figure 3-19: Circular aperture and its radiation pattern.
Table 3-1: Radiation patterns produced by various types of amplitude distributions along x1 over a rectangular
aperture lx × ly .
Amplitude distributiona
Cosine:
E1 (x1 ) = cosn (π x1 /2)
n = 0e
Relative
directivityb
Sidelobe
level
(dB)c
Half-power
(−3 dB)
beamwidth
(radians)d
1.00
13.2
0.88λ /lx
0.81
0.67
0.58
0.52
23
32
40
48
1.20λ /lx
1.45λ /lx
1.66λ /lx
1.94λ /lx
1.00
0.99
0.97
0.83
13.2
15.8
17.1
20.6
0.88λ /lx
0.92λ /lx
0.97λ /lx
1.15λ /lx
0.75
26.4
1.28λ /lx
1
9
n = 1>
=
2
3>
;
4
−1
x1
+1
Parabolic:
E1 (x1 ) = 1 − (1 − ∆)x12
9
∆ = 1.0e >
=
0.8
0.5 >
;
0
1
∆
−1
x1
+1
Triangular:
E1 (x1 ) = 1 − |x1 |
1
−1
x1
+1
a The variable x = (2/l )x , and |x | ≤ 1. b Relative to a uniform aperture distribution. c Below maximum intensity. d In x–z plane. e Same
x a
1
1
as uniform-distribution case.
Table 3-2: Radiation patterns produced by the
amplitude distribution function E(r1 ) = (1 − r12 )n over
a circular aperture of diameter d.
n
0 (uniform)
1
2
3
Relative
directivitya
Sidelobe
level
(dB)b
Half-power
(−3 dB)
beamwidth
(radians)
1.00
0.75
0.55
0.45
17.6
24.6
30.6
36.1
1.02λ /d
1.27λ /d
1.47λ /d
1.65λ /d
The variable r1 = (2/d)ra and 0 ≤ r1 ≤ 1. a Relative to a
uniform aperture distribution. b Below maximum intensity.
F(γ)
0 dB
z Q = (R, θ, ϕ)
θ ya
l
xa
–3
–5
Fu(γ)
l
Fc(γ)
–10
βxz
–13.2 dB
–15
–20
–23 dB
–25
–3
–2
–1
–30
1 1.5 2
γ = (lx/λ) sin θ
3
Figure 3-20: Comparison of the antenna pattern Fu (γ )
of a uniformly illuminated aperture with Fc (γ ) for a
cosine-tapered illuminated aperture.
Phase
Amplifiers Antenna
shifters (or attenuators) elements
ψN–1
aN–1
ψN–2
aN–2
ψi
ai
ψ1
a1
ψ0
a0
(a) Array elements with individual
amplitude and phase control
z
Q = (R0, θ, ϕ)
RN–1
d
(N – 1)d
d
Element N – 1
Element N – 2
Element i
Element 1 θ
Element 0
Ri
R0
y
(b) Array geometry relative to
observation point
Figure 3-21: Linear-array configuration and geometry.
z
Q
RN–1
d
Element N − 1 zN–1
Element N − 2 zN–2
Element i
id
≈
zi
R1
R0
Element 1
θ
id cos θ
}
Element 0
y
Figure 3-22: The rays between the elements and a faraway observation point are approximately parallel lines.
Hence, the distance Ri ≈ R0 − id cos θ .
N=7
d = λ/2
z
N=7
d = 3λ/2
0 dB
–3 dB
–10
θ
6d
0 dB
–3 dB
–10
–20
1
1
1
1
1
1
1
z
θ
–20
–30
6d
1
1
1
1
1
1
1
–30
(b) Uniform distribution, d = 3λ/2
(a) Uniform distribution, d = λ/2
N=7
d = λ/2
z
0 dB
θ
–5
–10
6d
1
6
15
20
15
6
1
–20
25˚
(c) Binomial distribution
Figure 3-23: Normalized array patterns of 7-element arrays: (a) uniform distribution with d = λ /2, (b) uniform distribution
with d = 3λ /2, and (c) binomial distribution with d = λ /2.
z
–(N – 1)δ
aN–1
N–1
–(N – 2)δ
aN–2
N–2
–iδ
ai
i
–2δ
a2
2
–δ
a1
1
θ
a0
Q
R0
y
0
Phase
shifters Attenuators
Figure 3-24: The application of linear phase.
End-fire θ0 = 0°
0 dB
θ0
°
45
–20
=
N = 10
d = λ/2
θ
Fan(θ)
.5°
15
48.7°
B
d
–10
–3
–30
10.2°
10.2
θ0 = 90°
Figure 3-25: Normalized array pattern of a 10-element
array with λ /2 spacing between adjacent elements. All
elements are excited with equal amplitude. Through
the application of linear phase across the array, the
main beam can be steered from the broadside direction
(θ0 = 90◦ ) to any scan angle θ0 . Equiphase excitation
corresponds to θ0 = 90◦ .
E
(a) Pyramidal
E
(b) E-plane sectoral
E
(c) H-plane sectoral
(d) Conical
Figure 3-26: Commonly used types of horn antennas.
Parabolic
main reflector
Main reflector
Horn feed
Primary
horn feed
(a) Front feed
paraboloid
Hyperbolic
subreflector
(b) Cassegrain
Parabolic
reflector section
Subreflector
Horn
(c) Horn-reflector
(d) Near-field Cassegrain
Figure 3-27: Horn-reflector antennas.
x
a
H
b
E
z
y
a
(a) Geometry
x = b/2
x
x=0
o
lo
z
θe
le
x = −b/2
(b) x-z Plane cross section
Figure 3-28: E-plane sectoral horn geometry and
coordinates.
x
H
b
E
z
y
(a) Geometry
y = −a/2
lh
y=0
o
lo
z
θh
y
y = −a/2
(b) y-z Plane cross section
Figure 3-29: H-plane sectoral horn geometry and
coordinates.
140
b
(λ/a)De
110
le/λ = 100
le
75
80
50
30
50
20
20
15
6 10
0
5
10
15
b/λ
20
25
30
Figure 3-30: Directivity of E-plane sectoral horns with
aperture height b and width a (based on Fig. 16.4 of
Schelkunoff and Friis, 1952).
140
lh/λ = 100
a
(λ/b)Dh
110
75
lh
50
80
30
20
15
10
8
6
50
20
0
5
10
15
a/λ
20
25
30
Figure 3-31: Directivity of H-plane horns of width a
and height b (based on Fig. 16.3 of Schelkunoff and Friis,
1952).
32
30
50
d
28
26
lo = 75
lc
30
lo
20
15
10
8
Directivity Dc (dB)
24
22
6
20
4
18
16
2
14
12
10
Optimum horn line
lo = 0.5 (maximum directivity
for a fixed length lo)
8
0.6 1.0
2
4 5 6 8 10
20
Diameter of horn aperture in wavelengths, d/λ
Figure 3-32: The directivity of a conical horn as a
function of axial length and aperture diameter.
Equiphase front
0
(a) Empty sectoral horn
Equiphase front
x = x1
x=0
0
Hyperbolic surface
Lens
(b) Lens-corrected sectoral horn
Figure 3-33: A hyperbolic lens of the appropriate index
of refraction can transform the phase front to a uniform
distribution across the aperture.
c
e
b
d
i
a
h
f
g
Figure 3-34: Various types of slots cut in the walls of
a rectangular waveguide. Slots c and f do not radiate,
because they do not interrupt the flow of surface current
(see Fig. 3-35).
Figure 3-35: The flow of surface currents in the walls
of a rectangular waveguide excited with a TE10 mode.
xa
Q
b
E
R
θ
z
a
ya
Figure 3-36: Rectangular slot with a tangential electric
field in the xˆ a direction.
λg
2
Radiation pattern
3λg
4
λg
2
Short circuit
x
y
a
Input
gi
xi
b
z
(a) Resonant slot array, with maximum
radiation in broadside direction ( y)
Input
g1
g2
gi
gN
Short
circuit
λg
3λg
2
4
(b) Transmission-line equivalent circuit
Figure 3-37: The resonant slot array.
d
x
Variable-frequency
input
θ0
a
b
z
Matched load
Figure 3-38: With the nonresonant slot array, the
main beam may be steered in direction by varying the
frequency of the input signal.
h
L
W
ε
Substrate
Ground plane
(a) Microstrip antenna
Patch
ε
Substrate
Ground plane
Coaxial line
(b) Side view with coaxial feed line
Figure 3-39: (a) Top view of a rectangular patch
antenna and (b) side view showing how it connects to
a coaxial transmission line.
Microstrip line
h
L
W
ε
Substrate
Ground plane
Figure 3-40: Microstrip patch antenna fed by a
microstrip transmission line.
F(θ)
0 dB
30
θ
−10
30
−20
60
60
−30
−40
90
90
0 −10 −20 −30 −40 dB −40 −30 −20 −10 0
Figure 3-41: Far-field patterns of a rectangular patch on
an infinite ground plane (Jackson, 2007).
100 Ω 100 Ω
200 Ω
100 Ω 100 Ω
200 Ω
200 Ω
100 Ω
200 Ω
100 Ω
Corporate feed
50 Ω input
Figure 3-42: Tapered lines to match 100-Ω patches to a
50-Ω line (R. E. Munson, 1974).
50
Zi = R + jX
40
R
Zi (Ω)
30
20
X
10
0
−10
−20
1.53
1.575
1.55
1.57
1.59
Frequency (GHz)
1.61
Figure 3-43: Input impedance Zi of a microstrip antenna with L = 6.255 cm, W = 9.383 cm, h = 0.1524 cm,
and εr = 2.2 (Jackson, 2007).
Bandwidth (%)
10
8
6
4
εr = 2.2
10.8
2
0
0
0.01
0.02
0.03
h/λ0
0.04
0.05
0.08
0.1
(a) Bandwidth
Radiation efficiency ξ (%)
100
2.2
80
60
εr = 10.8
40
20
0
0
0.02
0.04
0.06
h/λ0
(b) Radiation efficiency
Figure 3-44: (a) Bandwidth and radiation efficiency
versus the normalized substrate thickness for a moderatepermittivity substrate and a high-permittivity substrate
(Jackson, 2007).
Array
elements
Phase shifters
Array feed
line
T/R switch
HPA
LNA
Up converter
Down converter
(a) Conventional array
Array
elements
LNA
LNA
LNA
LNA
PA
PA
PA
PA
Array
feed
line
T/R switch
Up converter
Transmit/
receive
modules
Down converter
(b) Active array
Figure 3-45: Comparison of (a) conventional and (b)
active array system.
Power amplifier
Power
conditioning
HPA
Directional
coupler
Variable-gain
Phase
amplifier
shifter
Calibration
feedback
Radar
electronics
Switch or
circulator
Control logic
LNA
Figure 3-46: Design of an active transmit/receive module.
Antenna
element
Switch or
circulator
Array
elements
Phase shifters
Array feed
line
LNA
A/D
(a) Analog
Array
elements
LNAs
A/D
A/D
A/D
A/D
Digital summer
Multiple-beam
digital
beamformer
(b) Digital
Figure 3-47: Comparison of (a) analog and (b) digital
beamforming.
An example of one of the interactive modules available at the book website: mrs.eecs.umich.edu.