1. Antenna Theory Chapter.2.6.1~2.7 Antennas
Min-Beom Ko

2. Contents Antennas & RF Devices Lab.
2.Fundamental Parameters Of Antennas
2.6 Directivity
2.6.1 Directional Patterns
-Kraus Equation
-Tai and Pereira Equation
-Comparison of exact and approximate values
2.6.2 Omnidirectional Patterns
-McDonald Equation
-Pozar Equation
-Comparison of exact and approximate value
2.7 Numerical Techniques
Antennas & RF Devices Lab.

3. 2.6. Directivity Antennas & RF Devices Lab.
-Short review of directivity
-Definition
𝐷𝑖𝑟𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦= 𝑅𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 𝑖𝑛 𝑎 𝑔𝑖𝑣𝑒𝑛 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝐼𝑠𝑜𝑡𝑟𝑜𝑝𝑖𝑐 𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦
Isotropic
Radiation Intensity
(2-1)
-In mathematical form,
𝐷= 𝑈 𝑈 0 = 4𝜋𝑈 𝑃 𝑟𝑎𝑑 𝑊/𝑢𝑛𝑖𝑡 𝑠𝑜𝑙𝑖𝑑 𝑎𝑛𝑔𝑙𝑒 𝑊/𝑢𝑛𝑖𝑡 𝑠𝑜𝑙𝑖𝑑 𝑎𝑛𝑔𝑙𝑒 =(𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑙𝑒𝑠𝑠)
(2-2)
-Partial directivity of antenna
𝐷= 𝐷 𝜃 + 𝐷 𝜙
𝐷 𝜃 = 4𝜋 𝑈 𝜃 𝑃 𝑟𝑎𝑑 𝜃 + 𝑃 𝑟𝑎𝑑 𝜙
𝐷 𝜙 = 4𝜋 𝑈 𝜙 𝑃 𝑟𝑎𝑑 𝜃 + 𝑃 𝑟𝑎𝑑 𝜙
(2-3)
Figure 2.1 Definition of Directivity.
(2-4)
(2-5)
Antennas & RF Devices Lab.

4. 2.6. Directivity -To more general expressions
(2-6)
𝑈= 𝐵 0 𝐹 𝜃,𝜙 ≅ 1 2𝜂 [ 𝐸 𝜃 0 (𝜃,𝜙) 2 + | 𝐸 ∅ 0 (𝜃,𝜙)| 2 ]
(2-7)
𝑈 𝑚𝑎𝑥 = 𝐵 0 𝐹 𝜃,𝜙 | 𝑚𝑎𝑥 = 𝐵 0 𝐹 𝜃,𝜙 𝑚𝑎𝑥
𝑃 𝑟𝑎𝑑 = Ω 𝑈(𝜃,𝜙) 𝑑Ω= 𝐵 𝜋 0 𝜋 𝐹 𝜃,𝜙 𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙
(2-8)
𝐷 𝜃,𝜙 = 4𝜋𝐹 𝜃,𝜙 0 2𝜋 0 𝜋 𝐹 𝜃,𝜙 𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙
(2-9)
𝐷 0 = 4𝜋𝐹 𝜃,𝜙 𝑚𝑎𝑥 0 2𝜋 0 𝜋 𝐹 𝜃,𝜙 𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙
(2-10)
𝐷 0 = 4𝜋 0 2𝜋 0 𝜋 𝐹 𝜃,𝜙 𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙 𝐹 𝜃,𝜙 | 𝑚𝑎𝑥 = 4π Ω 𝐴
(2-11)
Figure 2.2 Normalized three-dimensional amplitude field pattern(in linear scale). Of a 10-element linear array antenna with a uniform spacing d=0.25𝜆 and progressive phase shift 𝛽=−0.6𝜋 between the elements
Antennas & RF Devices Lab.

5. 2.6. Directivity -Beam Solid Angle Antennas & RF Devices Lab.
𝐷 0 = 4𝜋 Ω 𝐴
(2-11)
Ω 𝐴 = 0 2𝜋 0 𝜋 𝐹 𝜃,𝜙 𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙 𝐹 𝜃,𝜙 | 𝑚𝑎𝑥
(2-12)
“The beam solid angle Ω 𝐴 is defined as the solid angle through which all the power of the antenna would flow if its radiation intensity is constant(and equal to the maximum value of 𝑈) for all angles within Ω 𝐴 ”
Figure 2.3 Beam solid angle (1)
Figure 2.4 Beam solid angle (2)
Antennas & RF Devices Lab.

6. 2.6.1. Directional Patterns -Kraus Equation Antennas & RF Devices Lab.
“For antennas with one narrow major lobe and very negligible minor lobes, the beam solid angle is approximately equal to the product of the half-power beamwidths in two perpendicular planes.”
-The beam solid angle has been approximated by
Figure 2.5 Beam solid angle for nonsymmetrical and symmetrical radiation patterns
Ω 𝐴 = 0 2𝜋 0 𝜋 𝐹 𝜃,𝜙 𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙 𝐹 𝜃,𝜙 | 𝑚𝑎𝑥 ≅ Θ 1𝑟 Θ 2𝑟
(2-13)
( Θ 1 :𝐻𝑃𝐵𝑊 𝑖𝑛 𝑜𝑛𝑒 𝑝𝑙𝑎𝑛𝑒 , Θ 2 :𝐻𝑃𝐵𝑊 𝑖𝑛 𝑎 𝑝𝑙𝑎𝑛𝑒 𝑎𝑡 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟)
-With this approximation, (2-11)can be approximated by
𝐷 0 ≅ 4𝜋 Θ 1𝑟 Θ 2𝑟
(2-14)
-(2-14) can be written as
𝐷 0 ≅ 4𝜋 180/𝜋 2 Θ 1𝑑 Θ 2𝑑 = 41,253 Θ 1𝑑 Θ 2𝑑
(2-15)
-For planar arrays, a better approximation to (2-15) is
𝐷 0 ≅ 32,400 Θ 1𝑑 Θ 2𝑑
(2-16)
Antennas & RF Devices Lab.

7. 2.6.1. Directional Patterns -Tai & Pereira Equation
“The directivity can be obtained by approximating an arithmetic mean expression of the directivity obtained from the E-plane and the H-plane, respectively.”
-The power pattern in two principal planes
-The total directivity can be derived from arithmetic-mean of 𝐷 1 and 𝐷 2
| 𝐸 𝜃 (𝜃,0)| 2 , 𝑡ℎ𝑒 𝐸−𝑝𝑙𝑎𝑛𝑒 𝑝𝑎𝑡𝑡𝑒𝑟𝑛
(2-17)
1 𝐷 = 1 2 ( 1 𝐷 𝐷 2 )
(2-22)
| 𝐸 𝜙 (𝜃, 𝜋 2 )| 2 , 𝑡ℎ𝑒 𝐻−𝑝𝑙𝑎𝑛𝑒 𝑝𝑎𝑡𝑡𝑒𝑟𝑛
(2-18)
-Directivity of an antenna with a rotationally symmetrical pattern | 𝐸 𝜃 (𝜃,0)| 2
-Finally, directivity can be approximated as follows
𝐷 1 = 𝐸 𝜃 𝑚𝑎𝑥 𝜋 | 𝐸 𝜃 (𝜃,0)| 2 𝑠𝑖𝑛𝜃𝑑𝜃
(2-19)
𝐷 0 ≅ 32 ln 2 Θ 1𝑟 2 + Θ 2𝑟 2 = Θ 1𝑟 2 + Θ 2𝑟 2
(2-23)
- Directivity of an antenna with a rotationally symmetrical pattern | 𝐸 𝜙 (𝜃, 𝜋 2 )| 2
-(2-23) can be written as
𝐷 2 = 𝐸 𝜙 𝑚𝑎𝑥 𝜋 | 𝐸 𝜙 (𝜃, 𝜋 2 )| 2 𝑠𝑖𝑛𝜃𝑑𝜃
(2-20)
𝐷 0 = 72,815 Θ 1𝑑 2 + Θ 2𝑑 2
(2-24)
-Approximate expression for 𝐷 1 and 𝐷 2 in terms of Θ 1𝑟 , Θ 2𝑟
(2-21)
𝐷 1 ≅16 ln 2 Θ 1𝑟 2
𝐷 2 ≅16 ln 2 Θ 2𝑟 2
( Θ 1𝑟 :𝐻𝑃𝐵𝑊 𝑜𝑓 𝐸−𝑝𝑙𝑎𝑛𝑒 , Θ 2𝑟 :𝐻𝑃𝐵𝑊 𝑜𝑓 𝐻−𝑝𝑙𝑎𝑛𝑒)
Antennas & RF Devices Lab.

8. 2.6.1. Comparison of Exact and Approximate Values of 𝐷 0
(2-25)
Figure 2.6 Comparison of exact and approximate values of directivity for directional 𝑈= 𝑐𝑜𝑠 𝑛 𝜃 power patterns
Table 2.1 Comparison of exact and approximate values of directivity for directional 𝑈= 𝑐𝑜𝑠 𝑛 𝜃 power patterns
-The error due to Tai & Pereira’s formula is always negative.
-The error due to Tai & Pereira’s formula monotonically decreases as n increases.
-The error due to Kraus’s formula is negative for small n and positive for large values of n(=5.497≅5.5).
-Kraus’s formula leads to small error for n<11.28 while Tai & Pereira’s leads to smaller error for n >11.28.
Antennas & RF Devices Lab.

9. 2.6.2. Omnidirectional Patterns
-McDonald Equation
-The function chosen to approximate the radiation pattern is
-The use of this approximation allows the expression for I to be
E 𝜓 = sin b𝜓 bψ
(2-25)
(2-31)
𝐼≈ 𝜋 𝑏 − 𝑏 2
(𝜓:measured from the broadside direction, i.e, 𝜓=0 is at right angles to the antenna axis.)
(b: determines the range associated with the number of minor lobes.(= 159 𝐻𝑃𝐵𝑊(𝑑𝑒𝑔𝑟𝑒𝑒𝑠) ))
-The numerical directivity 𝐷 𝑑 is given by
-Thus a combination of b, (2-31), and (2-26) gives
(2-26)
𝐷 𝑑 = 𝐼
𝐷 𝑑 ≅ 62 𝐻𝑃𝐵𝑊(𝑑𝑒𝑔𝑟𝑒𝑒𝑠)− (𝐻𝑃𝐵𝑊(𝑑𝑒𝑔𝑟𝑒𝑒𝑠)) 2
(2-32)
-Where
𝐼= 0 𝜋 2 𝑠𝑖𝑛 2 𝑏𝜓𝑐𝑜𝑠𝜓 (𝑏𝜓) 2 𝑑𝜓
(2-27)
-If isotropic antenna is selected as the reference antenna
-I can be evaluated as
𝐷 0 ≅ 101 𝐻𝑃𝐵𝑊(𝑑𝑒𝑔𝑟𝑒𝑒𝑠)− (𝐻𝑃𝐵𝑊(𝑑𝑒𝑔𝑟𝑒𝑒𝑠)) 2
(2-33)
(2-28)
𝐼= 2𝑏+1 4 𝑏 2 𝑆𝑖 𝑏 𝜋+ 2𝑏−1 4 𝑏 2 𝑆𝑖 𝑏− 1 2 𝜋− 1 2 𝑏 2 𝑆𝑖( 𝜋 2 )
-Where
(2-29)
𝑆𝑖 𝑥 = 0 𝑥 𝑠𝑖𝑛𝑡 𝑡 𝑑𝑡
𝑆𝑖 𝜋 2 =1.37
-Appropriate approximation for the first two sine integrals in(2.28)
(2-30)
𝑆𝑖 𝑥 ≈ 𝜋 2 − 𝑐𝑜𝑠𝑥 𝑥
(For antennas of moderate or high gain, b is much greater than unity.)
Antennas & RF Devices Lab.

10. 2.6.2. Omnidirectional Patterns
-Pozar Equation
Figure 2.7 Directivity versus elevation-plane half-power beamwidth, for an omnidirectional antenna. The open circles are data computed from the approximate curve-fit equation, given in Equation(2-35)
-Omnidirectional patterns can often be approximated by
(2-34)
𝑈= 𝑠𝑖𝑛 𝑛 𝜃 0≤𝜃≤𝜋 , 0≤𝜙≤2𝜋
-An approximate curve-fit equation to the curve Figure
2.7 is given by
(2-35)
𝐷 0 ≅− /𝐻𝑃𝐵𝑊(𝑑𝑒𝑔𝑟𝑒𝑒𝑠)
Antennas & RF Devices Lab.

11. 2.6.2. Comparison of Exact and Approximate Values of 𝐷 0
Figure 2.8 Omnidirectional patterns with and without minor lobes.
Figure 2.9 Comparison of exact and approximate values of directivity for omnidirectional 𝑈= 𝑠𝑖𝑛 𝑛 𝜃 power patterns
-Both equations are for omnidirectional pattern antennas.
-The McDonald’s approximation should be more accurate for omnidirectional patterns with minor lobes.
-The Pozar’s approximation should be more accurate for omnidirectional patterns without minor lobes.
-Figure 2.9 can be used for design purposes.
Antennas & RF Devices Lab.

12. 2.6.2. Comparison of Exact and Approximate Values of 𝐷 0
Problem : Design an antenna with omnidirectional amplitude pattern with a half-power beamwidth of 90 ∘ .
Express its radiation intensity by 𝑈= 𝑠𝑖𝑛 𝑛 𝜃. Determine the value of n and attempt to identify elements
that such a pattern. Determine the directivity of the antenna.
Solution :
Since the HPBW is 90 ∘ ,
The directivity based on McDonald’s Equation is
𝑈 𝜃= 45 ∘ =0.5= 𝑠𝑖𝑛 𝑛 45 ∘ = (0.707) 𝑛
𝐷 0 =− /90 =1.516=1.807𝑑𝐵 So
The directivity based on Pozar’s Equation is
𝐷 0 = − (90) 2 =1.4825=1.71𝑑𝐵
𝑛=2
The exact directivity is
The error rate of each equation is
P rad = 0 2𝜋 0 𝜋 𝑠𝑖𝑛 2 𝜃𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙= 8𝜋 3
McDonald’s Equation : 1.1%
Pozar’s Equation : -1.17%
D 0 = 4𝜋 8𝜋/3 = 3 2 =1.761𝑑𝐵
Antennas & RF Devices Lab.

13. 2.7. Numerical Techniques. Antennas & RF Devices Lab.
Instead of using approximate expressions of Kraus, Tai and Pereira, McDonald, or Pozar alternate and more accurate techniques may be desirable.
-If radiation intensity of a given antenna is separable,
-If the integrations in (2-39) cannot be performed analytically,
0 𝜋 𝑓 𝜃 𝑠𝑖𝑛𝜃𝑑𝜃= 𝑖=1 𝑁 𝑓 𝜃 𝑖 𝑠𝑖𝑛 Δ 𝜃 𝑖
(2-36)
𝑈= 𝐵 0 𝑓 𝜃 𝑔(𝜙)
(2-40)
-The directivity for such a system is given by,
-For N uniform divisions over the 𝜋 interval,
𝐷 0 = 4𝜋 𝑈 𝑚𝑎𝑥 𝑃 𝑟𝑎𝑑
(2-37)
(2-41)
Δ 𝜃 𝑖 = 𝜋 𝑁
-Where
-When 𝜃 𝑖 is taken at the trailing edge of each division
𝑃 𝑟𝑎𝑑 = 𝐵 𝜋 0 𝜋 𝑓 𝜃 𝑔 𝜙 𝑠𝑖𝑛𝑑𝜃 𝑑𝜙
(2-38)
(2-42)
𝜃 𝑖 =𝑖 𝜋 𝑁 , 𝑖=1,2,3,….,𝑁
-Which can also be written as,
-When 𝜃 𝑖 is selected at the middle of each division
𝑃 𝑟𝑎𝑑 = 𝐵 𝜋 𝑔(𝜙) 0 𝜋 𝑓 𝜃 𝑠𝑖𝑛𝑑𝜃 𝑑𝜙
(2-39)
(2-43)
𝜃 𝑖 = 𝜋 2𝑁 + 𝑖−1 𝜋 𝑁 , 𝑖=1,2,3,….,𝑁
Antennas & RF Devices Lab.

14. 2.7. Numerical Techniques. 𝐷 0 = 4𝜋 𝑈 𝑚𝑎𝑥 𝑃 𝑟𝑎𝑑
-Directivity is defined by
Figure 2.10 Digitization scheme of pattern in spherical coordinates
𝐷 0 = 4𝜋 𝑈 𝑚𝑎𝑥 𝑃 𝑟𝑎𝑑
(2-37)
-If 𝜃 and 𝜙 variations are separable,
𝑃 𝑟𝑎𝑑 = 𝐵 0 ( 𝜋 𝑁 )( 2𝜋 𝑀 ) 𝑗=1 𝑀 𝑔( 𝜙 𝑗 ) 𝑖=1 𝑁 𝑓 𝜃 𝑖 sin⁡ 𝜃 𝑖
(2-44)
-If 𝜃 and 𝜙 variations are not separable,
𝑃 𝑟𝑎𝑑 = 𝐵 0 ( 𝜋 𝑁 )( 2𝜋 𝑀 ) 𝑗=1 𝑀 𝑖=1 𝑁 𝐹 𝜃 𝑖 , 𝜙 𝑗 𝑠𝑖𝑛 𝜃 𝑖
(2-45)

15. Summery Reference Antennas & RF Devices Lab. Kraus Tai and Pereira
-An antenna directivity with one narrow major lobe and a negligible small lobe can be approximate as follows.
Kraus
Tai and Pereira
𝐷 0 ≅ 4𝜋 Θ 1𝑟 Θ 2𝑟
(2-14)
𝐷 0 ≅ 32 ln 2 Θ 1𝑟 2 + Θ 2𝑟 2 = Θ 1𝑟 2 + Θ 2𝑟 2
(2-23)
-An omnidirectional pattern antenna’s directivity can be approximate as follows.
McDonald
Pozar
𝐷 0 ≅ 101 𝐻𝑃𝐵𝑊(𝑑𝑒𝑔𝑟𝑒𝑒𝑠)− (𝐻𝑃𝐵𝑊(𝑑𝑒𝑔𝑟𝑒𝑒𝑠)) 2
(2-33)
𝐷 0 ≅− /𝐻𝑃𝐵𝑊(𝑑𝑒𝑔𝑟𝑒𝑒𝑠)
(2-35)
-Numerical techniques.
Separable 𝜃,𝜙
Not separable 𝜃,𝜙
𝑃 𝑟𝑎𝑑 = 𝐵 0 ( 𝜋 𝑁 )( 2𝜋 𝑀 ) 𝑗=1 𝑀 𝑔( 𝜙 𝑗 ) 𝑖=1 𝑁 𝑓 𝜃 𝑖 sin⁡ 𝜃 𝑖
(2-44)
𝑃 𝑟𝑎𝑑 = 𝐵 0 ( 𝜋 𝑁 )( 2𝜋 𝑀 ) 𝑗=1 𝑀 𝑖=1 𝑁 𝐹 𝜃 𝑖 , 𝜙 𝑗 𝑠𝑖𝑛 𝜃 𝑖
(2-45)
Reference
[1] C. A. Balanis, Antenna Theory: Analysis and Design, New York:Wiley, 1982.
[2]J. D. Kraus, Antennas, McGraw-Hill, New York, 1988
[3]C.-T. Tai and C. S. Pereira, “An Approximate Formula for Calculating the Directivity of an Antenna,” IEEE Trans. Antennas Propagat., Vol. AP-24, No. 2, pp. 235–236, March 1976
[4]N. A. McDonald, “Approximate Relationship Between Directivity and Beamwidth for Broadside Collinear Arrays,” IEEE Trans. Antennas
Propagat., Vol. AP-2,No. 2,pp. 340–341, March 1978
[5]D. M. Pozar, “Directivity of Omnidirectional Antennas,” IEEE Antennas Propagat. Mag. Vol. 35, No. 5, pp. 50–51, October 1993.
Antennas & RF Devices Lab.

16. Thank You
Antennas & RF Devices Lab.