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Antenna & Propagation Antenna Parameters
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Chapter 2 – Antenna
Parameters
1. Antenna Radiation
2. Figure of Merit
3. Efficiency
4. Radiation Pattern
5. Antenna Impedance
6. Friss formula
Antenna & Propagation Antenna Parameters
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Objective
To revise various antenna parameters from a system point of view:
• what they are
• how they are calculated
• where they can be used
Antenna & Propagation Antenna Parameters
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Prerequisites
• Units for an angle: radian vs degree
• Definition of solid angle W:
– steradians (sr)
– degrees2
• Power density and radiation intensity
• Radiated power from a Poynting vector
• Radiation from a Hertzian dipole
• Radiation from a magnetic current element
o180=π
2
2
deg41253180.44 =
=π
ππ
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Solid Angle: Steradian
r
Surface area dA
Solid angle dΩRadius r
Subtended angles dθ and dφ
Sphere
2r
dAd =Ω
2
2 sin
r
ddr φθθ=
φθθ ddsin=
x
z
πφθθπ π
4sin2
0 0==Ω ∫ ∫ ddSolid angle of a sphere, steradians
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Co-ordinate System
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Isotropic Antenna
• A theoretical point source radiating power equally in all
directions, 100% efficiency.
• Power Density: (W/m2)
• Directivity and Gain: D = G = 1 = 0 dBi
24 r
PS T
π=
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ANTENNA RADIATION
1. Radiation pattern
2. Radiation power density
3. Radiation intensity
4. Half-power beamwidth
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Cutting Plane
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Radiation Pattern
• Plot of |Eq| as a function of (q, f) at a fixed r, giving rise to a 3-
dimensional radiation pattern.
• E-plane pattern:
– plot of |Eq| vs q (f = constant, e.g. f = 0)
• H-plane pattern:
– plot of |Eq| vs f (q = constant, e.g. q = p/2)
• Distinguish between:
– polar and rectangular (Cartesian) plots
– linear (ratio) and logarithmic (dB) scales
– main lobe (main beam), side lobes (minor lobes), nulls
– HPBW (qHP, fHP) and FNBW (qFN, fFN): qHP ~ qFN/2
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Polar Pattern
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Rectangular Pattern
Beam Efficiency (BE):
>90% for radiometry,
astronomy, radar, etc. rx_or_tx_power_total
FNBW_within_power_rx_or_txBE =
Lobes, nulls, and beamwidths (HPBW and FNBW)
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Radiation Properties
Main lobe
Half-power
Beamwidth
HPBW
Side Lobe
Back
lobe
Minor
lobes
First-nulls
Beamwidth
FNBW
Nulls
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E-Plane and H-Plane
r
eIdzjE
rjβ
θθβ −
=)(sin30
θθβ
θ sinsin30
Kr
IdzE ==
where K is a constant,
usually normalize K=1.
E-plane H-plane
z
x
θ
x
y
φ
Hertzian Dipole
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FIGURE OF MERIT
1. Directivity
2. Gain
3. Antenna reciprocity
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Gains: D, G
• Directive Gain: D(q,f) or GD(q,f)
• Directivity:D
• Power Gain: G or GP
• Relationship between D and G:
• Relationship between D and HPBW:
DG η=
Antenna D (ratio) D (dB)
isotropic 1 0
Hertzian dipole 1.5 1.76
λ/2 dipole 1.64 2.15η = antenna efficiency
D and G are power ratios.
HPHPHP
Dφθππ.
44=
Ω=
o
HP
o
HPHP
Dφθ
ππ
.
41253
180.4
2deg
2
=Ω
=
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EIRP and ERP
• Effective Isotropic Radiated Power (EIRP):
• Effective Radiated Power (ERP):
TT GPEIRP .=
dipoleG
EIRPERP
2/λ
=
dBdBmdBm EIRPERP 15.2−=
dBdBmdBm ERPEIRP 15.2+=
EIRP is preferred in theoretical treatment for its simplicity.
ERP is sometimes used in practice as λ/2 dipole is readily available.
64.1
..
2/
TT
dipole
TT GP
G
GPERP ==
λ
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Effective Area
• The effective area of an antenna can be used to calculate the received
power
• Q1. A thin (diameter = 5 mm) half-wave dipole at 900 MHz (mobile phone)
has Gain = 1.64.
– How long is the antenna? l = = m = mm
– Calculate its effective area. Ae= = m2 = mm2
– Assume a rectangle with the area Ae. If one side of the rectangle equals
the antenna length, what is the dimension of the other side? Compare
this with the antenna diameter!
• Q2. What is the effective area of an isotropic antenna?
17
SAtyPowerDensiAP eeR .. == GAe πλ4
2
=where
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Effective Length or
Effective Height• For an antenna receiving a far-field wave, the effective length
le is defined by:
– where
• Voc = open-circuit voltage at antenna terminals
• Ei = incident electric field
• le = effective length
• For an electrically short dipole, le = lphysical/2
18
e
i
OC l.EV =
Antenna & Propagation Antenna Parameters
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Reciprocity Theorem
• The Lorentz Reciprocity Theorem (which states that the E
and H fields generated by two different sources at the same
frequency must satisfy certain conditions) can be applied to
antennas to show that:
the receiving and transmitting
properties of an antenna can be
interchanged.• For example, the radiation pattern, feed-point impedance,
and gain are the same when a wire dipole antenna is used for
either transmitting or receiving.
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EFFICIENCY
1. Antenna efficiency
2. Radiation efficiency
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Radiation Resistance: Rr
• Rr is a fictitious resistance to account for the power radiated
by an antenna.
• Power radiated, (E,H = rms value)
• (1)
•
• But, according to (I = rms value) (2)
• Therefore, equating (1) & (2)
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∫∫= dSExHPr .
∫∫=ππ
φθθπ
0
222
0
sin 120
ddrE
Pr
∫∫=ππ
φθθπ
0
222
0
2 sin
120
1ddr
E
IRr
rr RIP 2=
Antenna & Propagation Antenna Parameters
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Radiation Resistance of an
Electrically Short Dipole
• Electrically short means dl < l/10
• Hertzian dipole is an electrically short dipole
• For example:
– If dl/l = 0.01, then Rr = 0.08 W
– If dl/l = 0.1, then Rr = 7.90 W
22
22
2 79080
=
=λλ
πdldl
Rr(Ω)
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Feed-Point Impedance: Za
• Za = antenna impedance at its feed-point.
• Za is complex generally.
• Za can be determined by numerical methods, such as
Moments Method, FDTD, etc.
• For a dipole with total l<l/2 (or monopole with h<l/4), Xa is
negative (i.e. capacitive).
23
aaa jXRZ +=Za
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Antenna Efficiency: η
• Definition:
• In a lossless antenna,
• In an antenna with loss,
• Losses are due to:
– conduction (ohmic) loss
– dielectric loss
– reflection (mismatch)
• Total efficiency:
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lossr
r
RR
R
losspowerradiatedpower
radiatedpower
+=
+=
η
ra RR =
lossra RRR +=
)1(2
Γ−== dcrdc ηηηηηη
Antenna & Propagation Antenna Parameters
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ANTENNA IMPEDANCE
1. Input Impedance
2. Bandwidth
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Feed-Point Impedance: Za
• Za = antenna impedance at its feed-point.
• Za is complex generally.
• Za can be determined by numerical methods, such as
Moments Method, FDTD, etc.
• For a dipole with total l<l/2 (or monopole with h<l/4), Xa is
negative (i.e. capacitive).
26
aaa jXRZ +=Za
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Dipole Antenna
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Feed-Point Impedance of a
Dipole Antenna• Dipole Za represented by a 4-element equivalent circuit.
• L, C, and R elements are determined from the physical dimensions of dipole, independent of frequency.
• <10% error in Ra and Xa for dipole length (2h) up to 0.6λ.
• Equivalent circuit can be used in electronic circuit simulation software such as PSPICE.
h = dipole half-length
a = dipole radius
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Equivalent Circuit
[Tang, Tieng & Gunn, IEEE Trans.AP, Jan. 1993, pp.100-103]
pFah
hc
7245.0)/2log(
0674.1231 −
=
[ ]pF
ahhc
−−
= 02541.0861.0)/2log(
89075.02
8006.032
[ ] 02389.02
31 )/2(40754.7)/2log(41288.0 −+= ahahR
[ ] HahhL µ6188.0)/2log(4813.12.0012.1
31 −=
Ω− k27408.7
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ANTENNA BANDWIDTH
1. Operating Frequency
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Antenna Bandwidth:
Frequency Range
• (There are various definitions of antenna bandwidth.)
• The bandwidth of an antenna is the band of frequencies, over which it is considered to perform acceptably.
• The wider the range of frequencies a band encompasses, the wider the bandwidth of the antenna.
• Antennas are ordered pre-tuned by the manufacturer, for use in a specified band segment.
• The trade-off in designing an antenna for a wider bandwidth is that it would generally not have as good of performance in comparison to a similar antenna that is optimized for a narrower bandwidth.
Antenna & Propagation Antenna Parameters
32Rick Graziani graziani@cabrillo.edu 32
Antenna Bandwidth:
Frequency Range
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FRIIS FORMULA
1. Definition
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Definition
The Friis Transmission Equation is used to calculate the power
received from one antenna (with gain G1), when transmitted
from another antenna (with gain G2), separated by a distance R,
and operating at frequency f or wavelength lambda.
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Frii’s Transmission Equation
• Example:
– For a transmitter-receiver distance of 20 m at 10.2 GHz,
with PT = 100 W, PR = 3 mW, and GR = 15 dB, (a) what is
the required GT? (b) what is the maximum power density
at a point 20 m from the transmitter?
– Ans. (a) 18.4 dB, (b) 1.44 W/m2 or 0.144 mW/cm2
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24 r
GPS TT
π=
πλ
π 4.
4.
2
2
RTTeR
G
r
GPASP == RT
T
R GGrP
P2
4
=πλ
S = power density
Ae = effective area
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Simple Questions• What is the difference between POWER GAIN, DIRECTIVE
GAIN, and DIRECTIVITY of an antenna?
• What is the difference between EIRP and ERP?
• What is the significance of the effective area of a receiving
antenna?
• What is the significance of the radiation resistance of a
transmitting antenna?
• How would the feed-point impedance (sometimes called the
antenna impedance) affect the performance of an antenna?
• Derive the Frii’s Transmission Equation and/or the Radar
Equation.
• What is the significance of the G/T ratio of a satellite
receiving station?36