1. Antenna Theory Chapter.4.7.4~4.8.1 Antennas
Min-Beom Ko

2. Contents 4.7.4 Antennas for Mobile Communication Systems
4.7.5 Horizontal Electric Dipole
4.8 Ground Effects
4.8.1 Vertical Electric Dipole
Antennas & RF Devices Lab.

3. 4.7.4 Antennas for Mobile Communication Systems
Dipole
Monopole
L-Monopole
-The dipole and monopole are two of the most widely used antennas for wireless mobile communication systems.
Antennas & RF Devices Lab.

4. 4.7.4 Antennas for Mobile Communication Systems
<Inverted F Antenna>
<Loop Antenna>
<PIFA Antenna>
<Helix antenna>
-An alternative to the monopole for the handheld unit is loop. Other elements include the inverted F, planar inverted F antenna(PIFA), microstrip, spiral and others.
Antennas & RF Devices Lab.

5. 4.7.4 Antennas for Mobile Communication Systems
Figure Input impedance, real and imaginary parts, of a vertical monopole mounted on an experimental cellular telephone device
Figure Input impedance, real and imaginary parts, of a wire folded loop mounted vertically on an experimental cellular telephone device
-The position of the monopole element on the unit influences the pattern while it does not strongly affect the input impedance and resonant frequency.
-The order of the types of resonance(series vs parallel) can be interchanged by choosing another element.
Antennas & RF Devices Lab.

6. 4.7.4 Antennas for Mobile Communication Systems
Antennas & RF Devices Lab.

7. 4.7.4 Antennas for Mobile Communication Systems
Figure 4.23 Triangular array of dipoles used as a sectoral base-station antenna for mobile communication.
-It is a triangular array configuration consisting of twelve dipoles, with four dipoles on each side of the triangle.
-Each four-element array, on each side of the triangle, is used to cover an angular sector of 120 ° , forming what is usually referred to as a sectoral array.
Antennas & RF Devices Lab.

8. 4.7.5 Horizontal Electric Dipole
Figure 4.24 Horizontal electric dipole, and its associated image, above an infinite, flat, perfect electric conductor.
𝑬 𝝍 𝒅 =𝒋𝜼 𝒌 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌 𝒓 𝟏 𝟒𝝅 𝒓 𝟏 𝒔𝒊𝒏𝝍
(4-111)
𝑬 𝝍 𝒓 =𝒋𝜼 𝑹 𝒉 𝒌 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌 𝒓 𝟐 𝟒𝝅 𝒓 𝟐 𝒔𝒊𝒏𝝍
(4-112)
𝑬 𝝍 𝒓 =𝒋𝜼(−𝟏) 𝒌 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌 𝒓 𝟐 𝟒𝝅 𝒓 𝟐 𝒔𝒊𝒏𝝍
(4-112a)
<Reflection Coefficient 𝑹 𝒉 >
<Parallel Polarization- for 𝝓=𝟗 𝟎 𝝄 ,𝟐𝟕 𝟎 𝝄 >
𝑹 || = 𝜼 𝟏 𝒄𝒐𝒔 𝜽 𝒕 − 𝜼 𝟎 𝒄𝒐𝒔 𝜽 𝒊 𝜼 𝟏 𝒄𝒐𝒔 𝜽 𝒕 + 𝜼 𝟎 𝒄𝒐𝒔 𝜽 𝒊
𝚻= 𝟐 𝜼 𝟏 𝒄𝒐𝒔 𝜽 𝒕 𝜼 𝟎 𝒄𝒐𝒔 𝜽 𝒊 + 𝜼 𝟏 𝒄𝒐𝒔 𝜽 𝒕
<Perpendicular Polarization- for 𝝓= 𝟎 𝝄 ,𝟏𝟖 𝟎 𝝄 >
𝑹 ⊥ = 𝜼 𝟏 𝒄𝒐𝒔 𝜽 𝒊 − 𝜼 𝟎 𝒄𝒐𝒔 𝜽 𝒕 𝜼 𝟏 𝒄𝒐𝒔 𝜽 𝒊 + 𝜼 𝟎 𝒄𝒐𝒔 𝜽 𝒕
𝚻= 𝟐 𝜼 𝟏 𝒄𝒐𝒔 𝜽 𝒊 𝜼 𝟏 𝒄𝒐𝒔 𝜽 𝒊 + 𝜼 𝟎 𝒄𝒐𝒔 𝜽 𝒕
𝜼 𝟎 = 𝝁 𝟎 𝜺 𝟎
𝜼 𝟏 = 𝒋𝝎 𝝁 𝟐 𝝈 𝟐 +𝒋𝝎 𝜺 𝟐
Figure 1.13 Geometry for a plane wave obliquely incident at the interface between two dielectric regions
Antennas & RF Devices Lab.

9. 4.7.5 Horizontal Electric Dipole
<angle measured from the y-axis toward the observation point>
𝒄𝒐𝒔 𝝍 = 𝒂 𝒚 ∙ 𝒂 𝒓 = 𝒂 𝒚 ∙ 𝒂 𝒙 𝒔𝒊𝒏𝜽𝒄𝒐𝒔𝝓+ 𝒂 𝒚 𝒔𝒊𝒏𝜽𝒄𝒐𝒔𝝓+ 𝒂 𝒛 𝒄𝒐𝒔𝜽 =𝒔𝒊𝒏𝜽𝒔𝒊𝒏𝝓
(4-113)
𝒔𝒊𝒏𝝍= 𝟏− 𝒄𝒐𝒔 𝟐 𝝍 = 𝟏− 𝒔𝒊𝒏 𝟐 𝜽 𝒔𝒊𝒏 𝟐 𝝓
(4-114)
Since for far-field observations
𝒓 𝟏 ≅𝒓−𝒉𝒄𝒐𝒔𝜽 𝒓 𝟐 ≅𝒓+𝒉𝒄𝒐𝒔𝜽
(4-115a)
For phase variations
𝒓 𝟏 ≅ 𝒓 𝟐 ≅𝒓
For amplitude variations
(4-115b)
Total field, which is valid only above the ground plane
𝑬 𝝍 = 𝑬 𝝍 𝒅 + 𝑬 𝝍 𝒓 =𝒋𝜼 𝒌 𝑰 𝟎 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓 𝟏− 𝒔𝒊𝒏 𝟐 𝜽 𝒔𝒊𝒏 𝟐 𝝓 𝟐𝒋𝒔𝒊𝒏(𝒌𝒉𝒄𝒐𝒔𝜽)
(4-116)
(ㅡ:The field of single element , ㅡ:Array factor )
Figure 4.25 Horizontal electric dipole above an infinite perfect electric conductor
Antennas & RF Devices Lab.

10. Example 4.5
Using the vector potential 𝐀 and the derive the far zone spherical electric and magnetic field components of a horizontal infinitesimal dipole placed at the origin of the coordinate system of Figure 4.1.
Solution: Using (4-4), but for a horizontal infinitesimal dipole of uniform current directed along the y-axis, the corresponding vector potential can be written as
𝑨= 𝒂 𝒚 𝝁 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓
Using rectangular to spherical components transformation of (4-5), expressed as
𝑨 𝜽 = 𝑨 𝒚 𝒄𝒐𝒔 𝜽 𝒔𝒊𝒏 𝝓 = 𝝁 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓 𝒄𝒐𝒔 𝜽 𝒔𝒊𝒏 𝝓
𝑨 𝒓 𝑨 𝜽 𝑨 𝝓 = 𝒔𝒊𝒏 𝜽 𝒄𝒐𝒔 𝝓 𝒔𝒊𝒏 𝜽 𝒔𝒊𝒏 𝝓 𝒄𝒐𝒔 𝜽 𝒄𝒐𝒔 𝜽 𝒔𝒊𝒏 𝝓 𝒄𝒐𝒔 𝜽 𝒔𝒊𝒏 𝝓 − 𝒔𝒊𝒏 𝜽 − 𝒔𝒊𝒏 𝝓 𝒄𝒐𝒔 𝝓 𝟎 𝑨 𝒙 𝑨 𝒚 𝑨 𝒛
𝑨 𝝓 = 𝑨 𝒚 𝒄𝒐𝒔 𝝓 = 𝝁 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓 𝒄𝒐𝒔 𝝓
Using (3-58a) and (3-58b), we can write the corresponding far-zone electric and magnetic field components as
𝑬 𝜽 ≅−𝒋𝝎 𝑨 𝜽 =−𝒋 𝒘𝝎𝝁 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓 𝒄𝒐𝒔 𝜽 𝒔𝒊𝒏 𝝓
𝑯 𝜽 ≅− 𝑬 𝝓 𝜼 =−𝒋 𝒘𝝎𝝁 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝜼𝒓 𝒄𝒐𝒔 𝝓
𝑬 𝝓 ≅−𝒋𝝎 𝑨 𝝓 =−𝒋 𝒘𝝎𝝁 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓 𝒄𝒐𝒔 𝝓
𝑯 𝝓 ≅− 𝑬 𝜽 𝜼 =−𝒋 𝒘𝝎𝝁 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝜼𝒓 𝒄𝒐𝒔 𝜽 𝒔𝒊𝒏 𝝓

11. 4.7.5 Horizontal Electric Dipole
𝑬 𝝍 = 𝑬 𝝍 𝒅 + 𝑬 𝝍 𝒓 =𝒋𝜼 𝒌 𝑰 𝟎 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓 𝟏− 𝒔𝒊𝒏 𝟐 𝜽 𝒔𝒊𝒏 𝟐 𝝓 𝟐𝒋𝒔𝒊𝒏(𝒌𝒉𝒄𝒐𝒔𝜽)
(4-116)
𝒋𝜼 𝒌 𝑰 𝟎 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓 𝟏− 𝒔𝒊𝒏 𝟐 𝜽 𝒔𝒊𝒏 𝟐 𝝓
: The field of a single isolated element
𝟐𝒋𝒔𝒊𝒏(𝒌𝒉𝒄𝒐𝒔𝜽)
: Array factor
𝑬 𝒕 = 𝑬 𝟏 + 𝑬 𝟐 = 𝒂 𝜽 𝒋𝜼 𝒌 𝑰 𝟎 𝒍 𝟒𝝅 𝒆 −𝒋 𝒌 𝒓 𝟏 − 𝜷/𝟐 𝒓 𝟏 𝒄𝒐𝒔 𝜽 𝟏 + 𝒆 −𝒋 𝒌 𝒓 𝟐 − 𝜷/𝟐 𝒓 𝟐 𝒄𝒐𝒔 𝜽 𝟐
(6-1)
𝜽 𝟏 ≅ 𝜽 𝟐 ≅𝜽
(6-2a)
𝒓 𝟏 ≅𝒓− 𝒅 𝟐 𝒄𝒐𝒔𝜽 𝒓 𝟐 ≅𝒓+ 𝒅 𝟐 𝒄𝒐𝒔𝜽
For phase variations
(6-2b)
𝒓 𝟏 ≅ 𝒓 𝟐 ≅𝒓
For amplitude variations
(6-2c)
𝑬 𝒕 = 𝒂 𝜽 𝒋𝜼 𝒌 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓 𝒄𝒐𝒔𝜽 𝒆 +𝒋 𝒌 𝒓 𝟏 + 𝜷/𝟐 + 𝒆 −𝒋 𝒌 𝒓 𝟏 + 𝜷/𝟐
(6-3)
𝑬 𝒕 = 𝒂 𝜽 𝒋𝜼 𝒌 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓 𝒄𝒐𝒔𝜽 𝟐𝒄𝒐𝒔 𝟏 𝟐 (𝒌𝒅𝒄𝒐𝒔𝜽+𝜷)
(6-3)
Figure 6.1 Geometry of a two-element array positioned along the z-axis
Antennas & RF Devices Lab.

12. 4.7.5 Horizontal Electric Dipole
Equation (4-116) and (6-3) is equal if 𝜙= 0 𝜊 and 𝛽= 180 𝜊
𝑬 𝝍 = 𝑬 𝝍 𝒅 + 𝑬 𝝍 𝒓 =𝒋𝜼 𝒌 𝑰 𝟎 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓 𝟏− 𝒔𝒊𝒏 𝟐 𝜽 𝒔𝒊𝒏 𝟐 𝝓 𝟐𝒋𝒔𝒊𝒏(𝒌𝒉𝒄𝒐𝒔𝜽)
(4-116) =
𝑬 𝒕 = 𝒂 𝜽 𝒋𝜼 𝒌 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓 𝒄𝒐𝒔𝜽 𝟐𝒄𝒐𝒔 𝟏 𝟐 (𝒌𝒅𝒄𝒐𝒔𝜽+𝜷)
(6-3)
Figure 6.3 Element, array factor, and total field patterns of a two-element array of infinitesimal horizontal dipoles with identical phase excitation (𝛽= 0 𝜊 , 𝑑=𝜆/4)
Figure 6.4 Pattern multiplication of element, array factor, and total array patterns of a two-element array of infinitesimal horizontal dipoles with 𝛽=+ 90 𝜊 , 𝑑=𝜆/4
Antennas & RF Devices Lab.

13. 4.7.5 Horizontal Electric Dipole
Figure 4.26 Elevation plane (𝜙= 90 𝜊 ) amplitude patterns of a horizontal infinitesimal electric dipole for different heights above an infinite perfect electric conductor
Figure 4.27 Three-dimensional amplitude pattern of an infinitesimal horizontal dipole a distance ℎ=𝜆 above an infinite perfect electric conductor
Figure 4.28 Elevation plane (𝜙= 90 𝜊 ) amplitude patterns of a horizontal infinitesimal electric dipole for heights 2𝜆 and 5𝜆 above an infinite perfect electric conductor
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑜𝑏𝑒𝑠 ≅2( ℎ 𝜆 )
The total number of lobes is equal to the integer that most closely is equal to 2 ℎ 𝜆 with unity being the smallest number
Antennas & RF Devices Lab.

14. Horizontal Half-wavelength Dipole -HFSS
ℎ=𝜆/8
ℎ=𝜆/4
ℎ=𝜆/2
ℎ=𝜆
ℎ=2𝜆

15. 4.7.5 Horizontal Electric Dipole
The radiated power can be written as
𝑷 𝒓𝒂𝒅 =𝜼 𝝅 𝟐 𝑰 𝟎 𝒍 𝝀 𝟐 𝟐 𝟑 − 𝒔𝒊𝒏 𝟐𝒌𝒉 𝟐𝒌𝒉 − 𝒄𝒐𝒔 𝟐𝒌𝒉 (𝟐𝒌 𝒉) 𝟐 + 𝒔𝒊𝒏 𝟐𝒌𝒉 𝟐𝒌𝒉 𝟑
(4-118)
And the radiation resistance as
The directivity can be written as
𝑹 𝒓 =𝜼 𝝅 𝟐 ( 𝒍 𝝀 ) 𝟐 𝟐 𝟑 − 𝒔𝒊𝒏 𝟐𝒌𝒉 𝟐𝒌𝒉 − 𝒄𝒐𝒔 𝟐𝒌𝒉 (𝟐𝒌 𝒉) 𝟐 + 𝒔𝒊𝒏 𝟐𝒌𝒉 𝟐𝒌𝒉 𝟑
𝑫 𝟎 = 𝟒𝝅 𝑼 𝒎𝒂𝒙 𝑷 𝒓𝒂𝒅 = 𝟒 𝒔𝒊𝒏 𝟐 𝒌𝒉 𝑹(𝒌𝒉) 𝟒 𝑹 𝒌𝒉
(4-119)
𝒌𝒉≤𝝅/𝟐(𝒉≤𝝀/𝟒)
(4-123a)
By expanding the sine and cosine functions into series,
(4-119) reduces for small values of 𝑘ℎ to
(4-123b)
𝒌𝒉>𝝅/𝟐(𝒉>𝝀/𝟒)
Where 𝑅(𝑘ℎ) is
𝑹 𝒓 𝒌𝒉→𝟎 𝜼 𝝅 𝟐 ( 𝒍 𝝀 ) 𝟐 𝟐 𝟑 − 𝟐 𝟑 + 𝟖 𝟏𝟓 𝟐𝝅𝒉 𝝀 𝟐 =𝜼 𝟑𝟐 𝝅 𝟑 𝟏𝟓 𝒍 𝝀 𝟐 𝒉 𝝀 𝟐
(4-120)
𝑹 𝒌𝒉 = 𝟐 𝟑 − 𝒔𝒊𝒏 𝟐𝒌𝒉 𝟐𝒌𝒉 − 𝒄𝒐𝒔 𝟐𝒌𝒉 (𝟐𝒌 𝒉) 𝟐 + 𝒔𝒊𝒏 𝟐𝒌𝒉 𝟐𝒌𝒉 𝟑
(4-123c)
The radiation intensity is given by
For small values of 𝑘ℎ, (4-123a) reduces to
𝑼≅ 𝒓 𝟐 𝟐𝜼 𝑬 𝝍 𝟐 = 𝜼 𝟐 𝑰 𝟎 𝒍 𝝀 𝟐 𝟏− 𝒔𝒊𝒏 𝟐 𝜽 𝒔𝒊𝒏 𝟐 𝝓 𝒔𝒊𝒏 𝟐 (𝒌𝒉𝒄𝒐𝒔𝜽)
(4-121)
𝑫 𝟎 𝒌𝒉→𝟎 𝟒𝒔𝒊 𝒏 𝟐 𝒌𝒉 𝟐 𝟑 − 𝟐 𝟑 + 𝟖 𝟏𝟓 𝒌𝒉 𝟐 =𝟕.𝟓 𝒔𝒊𝒏 𝒌𝒉 𝒌𝒉 𝟐
(4-124)
The maximum value of (4-121) depends on the value of 𝑘ℎ
𝑼 𝒎𝒂𝒙 = 𝜼 𝟐 𝑰 𝟎 𝒍 𝝀 𝟐 𝒔𝒊𝒏 𝟐 (𝒌𝒉) 𝜼 𝟐 𝑰 𝟎 𝒍 𝝀 𝟐
𝒌𝒉≤𝝅/𝟐(𝒉≤𝝀/𝟒)
(4-122a)
(𝜽= 𝟎 𝝄 )
𝒌𝒉>𝝅/𝟐(𝒉>𝝀/𝟒)
𝝓= 𝟎 𝝄 𝒂𝒏𝒅 𝒔𝒊𝒏 𝒌𝒉𝒄𝒐𝒔 𝜽 𝒎𝒂𝒙 =𝟏
(4-122b)
𝒐𝒓 𝜽 𝒎𝒂𝒙 =𝒄𝒐 𝒔 −𝟏 (𝝅/𝟐𝒌𝒉)]
Antennas & RF Devices Lab.

16. 4.7.5 Horizontal Electric Dipole
𝑹 𝒓 =𝜼 𝝅 𝟐 ( 𝒍 𝝀 ) 𝟐 𝟐 𝟑 − 𝒔𝒊𝒏 𝟐𝒌𝒉 𝟐𝒌𝒉 − 𝒄𝒐𝒔 𝟐𝒌𝒉 (𝟐𝒌 𝒉) 𝟐 + 𝒔𝒊𝒏 𝟐𝒌𝒉 𝟐𝒌𝒉 𝟑
(4-119)
𝑫 𝟎 = 𝟒𝝅 𝑼 𝒎𝒂𝒙 𝑷 𝒓𝒂𝒅 = 𝟒 𝒔𝒊𝒏 𝟐 𝒌𝒉 𝑹(𝒌𝒉) 𝟒 𝑹 𝒌𝒉
𝒌𝒉≤𝝅/𝟐(𝒉≤𝝀/𝟒)
(4-123a)
𝒌𝒉>𝝅/𝟐(𝒉>𝝀/𝟒)
(4-123b)
𝑹 𝒌𝒉 = 𝟐 𝟑 − 𝒔𝒊𝒏 𝟐𝒌𝒉 𝟐𝒌𝒉 − 𝒄𝒐𝒔 𝟐𝒌𝒉 (𝟐𝒌 𝒉) 𝟐 + 𝒔𝒊𝒏 𝟐𝒌𝒉 𝟐𝒌𝒉 𝟑
(4-123c)
Figure 4.29 Radiation resistance and maximum directivity of a horizontal infinitesimal electric dipole as a function of its height above an infinite perfect electric conductor
Antennas & RF Devices Lab.

17. 4.7.5 Horizontal Electric Dipole
Figure 4.30 Input impedance of a horizontal 𝜆/2 above a flat lossy electric conducting surface
Figure 4.20 Input impedance of a vertical 𝜆/2 dipole above a flat lossy electric conducting surface.
-It is apparent that the conductivity does have a more pronounced effect on the impedance values, compared to those of the vertical dipole.
-As the height increases, the values of the resistance and reactance approach the values of the isolated element.
Antennas & RF Devices Lab.

18. 4.8 Ground Effects Antennas & RF Devices Lab.
Figure 4.14 Vertical electric dipole above infinite perfect electric conductor
Figure 4.25 Horizontal electric dipole above infinite perfect electric conductor
-In the previous two sections, the result is based on infinite PEC(𝜎=∞) ground that can not be realized.
-The analytical procedures that are introduced to examine the ground effects are based on the geometrical optics models of the previous sections. However, some additional details and approximations need to be added.
Antennas & RF Devices Lab.

19. 4.8 Ground Effects Antennas & RF Devices Lab.
1. The earth will be initially assumed to be flat.
- If radius of the earth is large compared to the wavelength and the observations angles are greater than about 57.3/ 𝑘𝑎 1/3 degrees from grazing ( a is earth radius)
2. Reflection coefficients are not unity and it will be a function of the angles of incidence and the constitutive parameters of the two media.
-For example in vertical electric dipole case, reflection coefficient is
𝑅 𝑣 = 𝜂 0 cos 𝜃 𝑖 − 𝜂 1 cos 𝜃 𝑡 𝜂 0 cos 𝜃 𝑖 + 𝜂 1 cos 𝜃 𝑡
3. Even though the spherical waves are radiated by the source, plane wave reflection coefficients are used.
4.If the height(h) is much less than the skin depth of the ground, the depth h below the interface should be increased by a complex distance 𝛿 1−𝑗 [𝛿= 2/(𝜔𝜇𝜎) ]
5.The characteristics of an antenna at low and medium frequencies are profoundly influenced by the lossy earth.
-this is particularly evident in the input resistance.
-When the antenna is located at a height that is small compared to the skin depth,
the input impedance may even be greater than its free-space values.
-improvements in the efficiency can be obtained by placing radial wires or metallic disks on the ground.
Figure 4.26 Image theory geometry
Antennas & RF Devices Lab.

20. 4.8.1 Vertical Electric Dipole
The reflection coefficient 𝑅 𝑣 is given by
𝑹 𝒗 = 𝜼 𝟎 𝒄𝒐𝒔 𝜽 𝒊 − 𝜼 𝟏 𝒄𝒐𝒔 𝜽 𝒕 𝜼 𝟎 𝒄𝒐𝒔 𝜽 𝒊 + 𝜼 𝟏 𝒄𝒐𝒔 𝜽 𝒕 =− 𝑹 ||
(4-125)
𝜂 0 = 𝜇 0 𝜀 0 =intrinsic impedance of free-space (air)
𝜂 1 = 𝑗𝜔𝜇 1 𝜎 1 +𝑗𝜔𝜀 1 =intrinsic impedance of the ground
𝜃 𝑖 =angle of the incidence (relative to the normal)
𝜃 𝑡 =angle of refraction (relative to the normal)
Snell’s law of refraction
𝜸 𝟎 𝒔𝒊𝒏 𝜽 𝒊 = 𝜸 𝟏 𝒔𝒊𝒏 𝜽 𝒕
(4-126)
𝛾 0 =𝑗 𝑘 0 = propagation constant for free-space (air)
𝑘 0 = phase constant for free-space (air)
𝛾 1 =( 𝛼 1 +𝑗 𝑘 1 )= propagation constant for the ground
𝛼 1 =attenuation constant for the ground
𝑘 1 =phase constant for the ground
Using the far-field approximations, the total electric field above the ground can be written as
𝑬 𝜽 =𝒋𝜼 𝒌 𝑰 𝟎 𝒍 𝒆 −𝒋𝒌𝒓 𝟒𝝅𝒓 𝒔𝒊𝒏𝜽 𝒆 𝒋𝒌𝒉𝒄𝒐𝒔 𝜽 + 𝑹 𝒗 𝒆 −𝒋𝒌𝒉𝒄𝒐𝒔𝜽
(4-127)
Figure 4.14 Vertical electric dipole above infinite perfect electric conductor

21. (di𝑒𝑙𝑒𝑡𝑟𝑖𝑐 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 : 𝜀 𝑟 5~100 , 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 : 𝜎( 10 −4 ~1 𝑆/𝑚))
4.8.1 Vertical Electric Dipole
Permittivity and conductivity of the earth are strong functions of the ground’s constituents, especially its moisture.
(di𝑒𝑙𝑒𝑡𝑟𝑖𝑐 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 : 𝜀 𝑟 5~100 , 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 : 𝜎( 10 −4 ~1 𝑆/𝑚))
𝜼 𝟏 = 𝒋𝝎𝝁 𝟏 𝝈 𝟏 +𝒋𝝎𝜺 𝟏
𝑹 𝒗 = 𝜼 𝟎 𝒄𝒐𝒔 𝜽 𝒊 − 𝜼 𝟏 𝒄𝒐𝒔 𝜽 𝒕 𝜼 𝟎 𝒄𝒐𝒔 𝜽 𝒊 + 𝜼 𝟏 𝒄𝒐𝒔 𝜽 𝒕 =− 𝑹 ||
Figure 4.31 Elevation plane amplitude patterns of an infinitesimal vertical dipole above a perfect electric conductor(𝜎=∞) and a flat earth
(𝜎=0.01𝑆/𝑚 , 𝜖 𝑟1 =5, 𝑓=1 𝐺𝐻𝑧 )
-Thus the ground effects on the pattern of a vertically polarized antenna are significantly different from those of a perfect conductor.
-Significant changes also occur in the impedance.

22. Thank You
Antennas & RF Devices Lab.