1. ANTENNA THEORY by Constantine A. Balanis Chapter 2.13 – 2.15.2
Yun-tae Park
Antennas & RF Devices Lab.

2. Contents 2. Fundamental Parameters of Antennas 2.5 Beamwidth
2.13 Input Impedance
2.14 Antenna Radiation Efficiency
2.15 Antenna Vector Effetive Length and Equivalent Areas
Vetor Effective Length
Antenna Equivalent Areas
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3. 2.5 Beamwidth
Beamwidth - The angular separation between two identical points on opposite side of
the pattern maximum.
Half-Power Beamwidth (HPBW) - In a plane containing the direction of the maximum of a beam, the angle between the two directions in which the radiation intensity is one-half value of the beam.
First-Null Beamwidth (FNBW) - The angular separation between the first nulls of the pattern.
Antennas & RF Devices Lab.

4. 2.5 Beamwidth
Beamwidth of the antenna is used to describe the resolution capabilities of the antenna to distinguish between two adjacent radiating sources or radar targets. -
distinguish -
difficult to distinguish
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5. - The impedance presented by an antenna at its terminals.
2.13 Input Impedance
Input impedance
- The impedance presented by an antenna at its terminals.
- The ratio of the voltage to current at a pair of terminals.
- The ratio of the appropriate components of the electric to magnetic fields at a point.
(2-72)
(2-73)
Figure Transmitting antenna and its equivalent circuits.
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6. Antennas & RF Devices Lab.
2.13 Input Impedance
(2-74)
(Transmitting mode)
Find the amount of power delivered to 𝑅 𝑟 for radiation and
amount of dissipated in 𝑅 𝐿 as heat.
Figure Transmitting antenna and its equivalent circuits.
(2-75)
(2-75a)
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7. the power delivered to the antenna for radiation
2.13 Input Impedance
the power delivered to the antenna for radiation
(2-76)
dissipated as heat
(2-77)
dissipated as heat on the internal resistance 𝑅 𝑔 of the generator
(2-78)
Figure Transmitting antenna and its equivalent circuits.
Maximum power delivered to the antenna when conjugate matching.
(2-79)
(2-80)
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8. Antennas & RF Devices Lab.
2.13 Input Impedance
(when conjugate matching, ( ))
(2-81)
(2-82)
(2-83)
Figure Transmitting antenna and its equivalent circuits.
(2-84)
(2-85)
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9. Antennas & RF Devices Lab.
2.13 Input Impedance
(receiving mode paralles that for the transmitting mode
under conjugate matching( ))
(2-86)
(2-87)
(2-88)
(2-89)
Figure Antenna and its equivalent circuits in the receiving mode.
Antennas & RF Devices Lab.

10. and the other half is scattered.
2.13 Input Impedance
(under conjugate matching, ( ) )
(2-89)
(2-87)
50%
50%
(2-86)
(2-88)
If the losses are zero ( 𝑅 𝐿 =0), then half of the captured power is delivered to the load
and the other half is scattered.
The most that can be delivered to the load is only half of that captured and that is only under conjugate matching and lossless transmission line.
Antennas & RF Devices Lab.

11. 2.14 Antenna Radiation Efficiency
The conduction and dielectric losses of an antenna are very difficult to compute.
lumped together to form the 𝑒 𝑐𝑑 efficiency
The resistance 𝑅 𝐿 is used to represent the conduction-dielectric losses.
conduction-dielectric efficiency ( 𝑒 𝑐𝑑 ) - the ratio of the power delivered to the radiation resistance 𝑅 𝑟
to the power delivered to 𝑅 𝑟 and 𝑅 𝐿
(2-76)
(2-90)
(2-77)
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12. 2.14 Antenna Radiation Efficiency
If the skin depth 𝛿 of the metal is very small compared to the smallest diagonal of the section of the rod, the current is confined to a thin layer near the conductor surface. Therefore the high-frequency resistance can be written, based on a uniform current distribution, as
(2-90b)
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13. 2.15 Antenna Vector Effective Length and Equivalent Areas
An antenna in the receiving mode is used to capture(collect) electromagnetic waves
and to extract power from them.
For each antenna, an equivalent length and a number of equivalent areas can be defined.
Equivalent quantities are used to describe the receiving characteristics of an antenna when a wave is incident upon the antenna.
Figure Uniform plane wave incident upon dipole and aperture antennas.
Antennas & RF Devices Lab.

14. 2.15 Antenna Vector Effective Length and Equivalent Areas
The effective length of an antenna is a quantity that is used to determine the voltage induced on the open-circuit terminals of the antenna when a wave impinges upon it.
(2-91)
It is a far-field quantity and it is related to the far-zone field 𝑬 𝑎 radiated by the antenna,
with current 𝑰 𝑖𝑛 in its terminals.
𝑰 𝑖𝑛
𝑬 𝑎
(2-92)
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15. 𝑬 𝑖 𝑬 𝑖 𝑬 𝑖 2.15 Antenna Vector Effective Length and Equivalent Areas
The effective length represents the antenna in its transmitting and receiving modes, and
it is particularly useful in relating the open-circuit voltage 𝑉 𝑜𝑐 of receiving antennas.
the effective length is a vector
𝑬 𝑖
𝑬 𝑖
when taking the maximum value over 𝜃,𝜙 this becomes
𝑬 𝑖
(2-93)
𝑬 𝑖
For linear antennas
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16. 2.15 Antenna Vector Effective Length and Equivalent Areas
Effective length of a linearly polarized antenna receiving a plane wave in a given direction
- The ratio of the magnitude of the open-circuit voltage developed at the terminals
of the antenna to the magnitude of the electric-field strength in the direction of the
antenna polarization.
- It is used to determine the polarization efficiency of the antenna.
Ex) a small dipole of length and with a triangular current distribution
𝑰 𝑖𝑛
𝑬 𝑎
(4-36a)
𝑰 𝑖𝑛
𝑬 𝑎
(2-92)
Figure 4.4 Geometrical arrangement of dipole and current distribution.
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17. 2.15 Antenna Vector Effective Length and Equivalent Areas
Antenna Equivalent Areas
Equivalent area - the power capturing characteristics of the antenna when a wave impinges on it
Effective area (aperture) - the ratio of the available power at the terminals of a receiving
antenna to the power flux density of a plane wave incident on the antenna
from that direction, the wave being polarization-matched to the antenna
(2-94)
Effective aperture - the area which when multiplied by the incident power density
gives the power delivered to the load.
(2-95)
Figure Equivalent circuits in the receiving mode.
Antennas & RF Devices Lab.

18. 2.15 Antenna Vector Effective Length and Equivalent Areas
Antenna Equivalent Areas
(2-95)
(when conjugate matching, ( ))
(2-96)
Scattering area - the equivalent area when multiplied by the incident power density is equal
to the scattered or reradiated power
(2-97)
Loss area - the equivalent area which when multiplied by the incident power density leads
to the power dissipated as heat through 𝑅 𝐿
(2-98)
Antennas & RF Devices Lab.

19. 2.15 Antenna Vector Effective Length and Equivalent Areas
Antenna Equivalent Areas
Capture area - the equivalent area which when multiplied by the incident power density leads
to the total power captured, collected, or intercepted by the antenna
(2-99)
Capture area = Effective area + Scattering area + Loss area
Aperture efficiency 𝜖 𝑎𝑝 - the ratio of the maximum effective area 𝐴 𝑒𝑚 of the antenna
to its physical area 𝐴 𝑝
𝜖 𝑎𝑝 = 𝐴 𝑒𝑚 𝐴 𝑝 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑎𝑟𝑒𝑎 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑎𝑟𝑒𝑎
(2-100)
𝐴 𝑒𝑚 ≤ 𝐴 𝑝 or ≤ 𝜖 𝑎𝑝 ≤ 1
Antennas & RF Devices Lab.

20. 2.15 Antenna Vector Effective Length and Equivalent Areas
Antenna Equivalent Areas
Partial effective area of an antenna for a given polarization in a given direction
- the ratio of the available power at the terminals of a receiving antenna to the power flux density of
a plane wave incident on the antenna from that direction and with a specified polarization
differing from the receiving polarization of the antenna
The effective area of an antenna is not necessarily the same as the physical aperture.
uniform amplitude and phase field distributions
nonuniform field distributions
The maximum effective area of wire antennas is greater than the physical area.
(if taken as the area of a cross section of the wire when split lengthwise along its diameter)
the wire antenna can capture much more power than is intercepted by its physical size
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21. 2.15 Antenna Vector Effective Length and Equivalent Areas
Antenna Equivalent Areas
Ex) a very short lossless dipole (𝑙≪𝜆) , the incident field is linearly polarized
along the axis of the dipole
Figure Uniform plane wave incident upon dipole antennas.
If , the effective area is only one-half of the maximum effective area given above.
Antennas & RF Devices Lab.

22. 2.15 Antenna Vector Effective Length and Equivalent Areas
Antenna Equivalent Areas
Figure Current distribution on linear dipoles.
Typical physical diameters (widths) of wires used for dipoles
the wire antenna can capture much more power than is intercepted by its physical size
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23. Thank you for your attention
Antennas & RF Devices Lab.