1. ANTENNA THEORY by Constantine A. Balanis Chapter 4.5.6 – 4.7.3
Yun-tae Park
Antennas & RF Devices Lab.

2. Contents 4. Linear Wire Antennas 4.5 Finite Length Dipole
4.5.6 Finite Feed Gap
4.6 Half-wavelength Dipole
4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.1 Image Theory
4.7.2 Vertical Electric Dipole
4.7.3 Approximate Formulas for Rapid Calculations and Design
Antennas & RF Devices Lab.

3. Antenna Measurement System
The three-dimensional plots of 𝑉 1𝑜𝑐 and 𝑉 2𝑜𝑐 ,
as a function of θ and 𝜙, field patterns
Due to reciprocity, the 3D graph of 𝑉 1𝑜𝑐 = 𝑉 2𝑜𝑐
the transmitting and receiving
field patterns are also equal
Figure 3.5 Antenna arrangement for pattern measurements and reciprocity theorem.
#1 the antenna under test
#2 the probe antenna - oriented to transmit or receive
maximum radiation
(a) antenna #1 is held stationary while #2 is allowed to move
on the surface of a constant radius sphere
(b) antenna #2 is maintained stationary
while #1 pivots about a point
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4. Antenna Measurement System
Horn antenna
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5. Fundamental mode of Patch Antenna
Figure Charge distribution and current density creation
on microstrip patch.
 propagation mode
height-to-width ratio is very small
Transverse electromagnetic (TEM) mode
- neither electric nor magnetic field in the direction of propagation
- most of the charge concentration and
current flow remain underneath the patch
- the tangential magnetic fields at the edges ≃0
- the fringing of the fields along the edges of the patch are also very small
- the electric field is nearly normal to the surface of the patch
Transverse electric (TE) mode
- no electric field in the direction of propagation
Transverse magnetic (TM) mode
- no magnetic field in the direction of propagation
Antennas & RF Devices Lab.

6. Fundamental mode of Patch Antenna
The mode with the lowest order resonant frequency is referred to
as the dominant mode.
Figure Rectangular microstrip patch geometry.
(14-21)
(14-23)
Figure Field configuration (modes) for rectangular microstrip patch.
Antennas & RF Devices Lab.

7. Antennas & RF Devices Lab.
4.5 Finite Length Dipole
4.5.6 Finite Feed Gap
nonzero current at the feed point for antennas with a finite gap at the terminals
(4-56)
p is a coefficient that is dependent upon the overall length of
the antenna and the gap spacing at the terminals.
The values of p become smaller as the radius of
the wire the gap decrease.
(4-81)
when 𝑙= 𝜆 2 ,
the shape of the current
not changed
(4-82)
when 𝑙=𝜆,
modified by the second term
which is more dominant for small values of 𝑧 ′
(4-83)
Antennas & RF Devices Lab.

8. Antennas & RF Devices Lab.
4.5 Finite Length Dipole
4.5.6 Finite Feed Gap
The current distribution based on the ideal current distribution is zero at the feed point; for practical antennas it is very small.
The gap at the feed plays an important role on the current distribution at and near the feed point.
Figure Current distribution on a dipole antenna.
Antennas & RF Devices Lab.

9. 4.6 Half-wavelength Dipole
Half-wavelength (l= λ 2 ) dipole
- most commonly used antenna
- radiation resistance : 73 ohms its matching to the line is simplified especially at resonance
(4-84)
(l= λ 2 )
(4-62)
(4-85)
▶ time-average power density
▶ radiation intensity
(4-86)
(4-87)
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10. 4.6 Half-wavelength Dipole
Figure 4.6 Elevation plane amplitude patterns for a thin
dipole with sinusoidal current distribution.
Figure Three-dimensional pattern of a 𝜆 2 dipole.
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11. 4.6 Half-wavelength Dipole
▶ total power radiated
(l= λ 2 )
(4-67)
(4-88)
(4-89)
▶ maximum directivity
(4-91)
Antennas & RF Devices Lab.

12. 4.6 Half-wavelength Dipole
▶ maximum effective area
▶ radiation resistance (𝜂⋍120𝜋)
(4-92)
(4-93)
the radiation resistance at the input terminals (input resistance)
The imaginary part (reactance) associated with the input impedance of a dipole is a function of its length.
(4-93a)
To reduce the imaginary part of the input impedance
- the antenna is matched
- reduced in length
Figure 4.8 Current distributions along the length of a linear wire antenna.
Antennas & RF Devices Lab.

13. 4.7 Linear Elements near or on Infinite Perfect Conductors
The presence of an obstacle, especially when it is near the radiating element,
can significantly alter the overall radiation properties of the antenna system.
the most common obstacle is the ground
The amount of reflected energy and its direction are
controlled by the geometry and constitutive parameters of the ground.
In general, the ground is a lossy medium (𝜎≠0) whose effective conductivity increases with frequency.
Assumed that the ground is a perfect electric conductor, flat, and infinite in extent.
Antennas & RF Devices Lab.

14. 4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.1 Image Theory
virtual sources (images) for the reflection
To analyze the performance of an antenna near an infinite plane conductor
These are not real sources but imaginary ones, which when combined with the real sources, form an equivalent system.
For an observation point 𝑃 1
- a direct wave
- a wave from the actual radiated toward point 𝑅 1
of the interface undergoes a reflection
For an observation point 𝑃 2
- a direct wave
- a wave from the actual radiated toward point 𝑅 2
of the interface undergoes a reflection
By extending its actual path below the interface,
it will seem to originate from a virtual source positioned a distance h below the boundary.
Figure Vertical electric dipole above an infinite, flat,
perfect electric conductor.
Antennas & RF Devices Lab.

15. 4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.1 Image Theory
The amount of reflection is generally determined by the respective constitutive parameters of the media below and above the interface.
For a perfect electric conductor below the interface
- incident wave is completely reflected
- the field below the boundary is zero
- tangential components of the electric field must vanish
along the interface (boundary condition)
For an incident electric field with vertical polarization, the polarization of the reflected waves must satisfy the boundary conditions.
The virtual source must also be vertical and with a polarity
in the same direction as that of the actual source.
Figure Vertical electric dipole above an infinite, flat,
perfect electric conductor.
Γ=1
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16. 4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.1 Image Theory
the radiating element in a horizontal position
Similar to that of the vertical dipole,
the virtual source (image) is also placed a distance h below the interface but with a 180° polarity difference relative to the actual source.
Γ=−1
Figure Horizontal electric dipole and its associated image,
above an infinite, flat, perfect electric conductor.
Antennas & RF Devices Lab.

17. 4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.1 Image Theory
Figure Field in PEC and image area.
Figure Field in PMC and image area.
( 𝐸 ⫽ is the parallel electric component in the PEC wall,
and 𝐻 ⊥ is the vertical magnetic component in the PEC wall.)
( 𝐸 ⊥ is the vertical electric component in the PMC wall,
and 𝐻 ⫽ is the parallel magnetic component in the PMC wall.)
Antennas & RF Devices Lab.

18. 4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.2 Vertical Electric Dipole
the fields of a vertical linear element near a perfect electric conductor
only far-field observations
the infinitesimal dipole
- length 𝑙
- constant current 𝐼 0
- observation point 𝑃
(4-94)
(4-95a)
the total field above the interface (𝑧≥0)
Figure Vertical electric dipole above infinite perfect electric conductor.
Since a field cannot exist inside a perfect electric conductor,
it is equal to zero below the interface.
Antennas & RF Devices Lab.

19. 4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.2 Vertical Electric Dipole
To simplify the expression for the total electric field,
For far-field observations (𝑟≫ℎ),
(4-97)
(4-98)
the total field
(4-99)
Figure Vertical electric dipole above infinite perfect electric conductor.
the field of a single source
array factor
Pattern multiplication
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20. 4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.2 Vertical Electric Dipole
The shape and amplitude of the field is not only controlled by the field of the single element
but also by the positioning of the element relative to the ground.
scalloping
(4-100)
Figure Elevation plane amplitude patterns of a vertical infinitesimal electric dipole for different heights above an infinite perfect electric conductor.
Figure Elevation plane amplitude patterns of a vertical infinitesimal
electric dipole for heights of 2𝜆 and 5𝜆 above an infinite perfect electric conductor.
Antennas & RF Devices Lab.

21. 4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.2 Vertical Electric Dipole
Since the total field of the antenna system is different from that of a single element, the directivity and radiation resistance are also different.
(4-103)
(4-101)
(4-103a)
(4-99)
(4-104)
(4-102)
Antennas & RF Devices Lab.

22. 4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.2 Vertical Electric Dipole
(4-105)
Figure Directivity and radiation resistance of a vertical infinitesimal electric dipole as a function of its height above an infinite perfect electric conductor.
Ex. Radiation resistance of an infinitesimal dipole (𝑙= 𝜆 50 )
Antennas & RF Devices Lab.

23. 4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.2 Vertical Electric Dipole
(4-84)
(4-85)
(4-106)
Figure Quarter-wavelength monopole on an infinite perfect electric conductor.
Antennas & RF Devices Lab.

24. 4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.2 Vertical Electric Dipole
It is apparent that the conductivity does not strongly influence
the impedance values.
It is observed that the values of the resistance and reactance approach,
as the height increases, the corresponding ones of the isolated element (73 ohms for the resistance and 42.5 ohms for the reactance).
Figure Input impedance of a vertical 𝜆 2 dipole above a flat lossy electric conducting surface.
Antennas & RF Devices Lab.

25. 4.7 Linear Elements near or on Infinite Perfect Conductors
4.7.3 Approximate Formulas for Rapid Calculations and Design
▶ input resistance of a dipole
(4-70)
(4-79)
▶ input resistance of monopole
(4-106)
using simpler but approximate expressions
Defining G as
(4-107a)
(4-107b)
(where l is the total length of each respective element)
Antennas & RF Devices Lab.

26. 4.7 Linear Elements Near or on Infinite Perfect Conductors
4.7.3 Approximate Formulas for Rapid Calculations and Design
simpler in form
much more convenient in design (synthesis) problems
- input resistance is given
- to determine the length of the element
(4-108a)
(4-108b)
Ex. The length of the dipole ( 𝑅 𝑖𝑛 =50 𝑜ℎ𝑚𝑠)
(4-109a)
(4-109b)
( 𝑅 𝑖𝑛 for 0.422𝜆 is ohms
𝑙 for 50 ohms is 𝜆)
(4-110a)
(4-110b)
Antennas & RF Devices Lab.

27. Thank you for your attention
Antennas & RF Devices Lab.