1. Antenna Theory
By CONSTANTINE A.BALANIS
Ch4.4.2~4.5.5
O Yeon Jeong

2. Contents 4. Linear Wire Antennas 4.8.2 Horizontal electric dipole
4.8.3 Earth curvature
5. Loop Antennas
Introduction
Small circular loop
5.2.1 Radiated fields
5.2.2 Small loop and infinitesimal magnetic dipole

3. Current distribution of thin wire
Figure. Linear dipole
Figure. A transmission line terminated in an open circuit.
Figure. Current variation along an open-circuited transmission line
z=− λ 2
z=− λ 4
z=0
z=0
z=− λ 4
z=− λ 8
z=−λ
z=−2 λ
z=0
Figure. Current distribution of λ 4 dipole
Figure. Current distribution of λ 2 dipole
Figure. Current distribution of λ dipole

4. Off-center fed dipole 𝐼 𝑜𝑓𝑓 𝑐𝑒𝑛𝑡𝑒𝑟−𝑓𝑒𝑑 𝐼 𝑜 z z 𝑅 𝑜𝑐𝑓 𝑅 𝑟
Figure1. center-fed dipole(symmetric)
Figure2. off center-fed dipole(asymmetric)

5. Finite diameter dipole
In practice, infinitely thin (electrically) wires are not realizable.
For a finite diameter wire (usually d > 0.05λ) the current distribution may not be sinusoidal and its effect on the radiation pattern of the antenna is usually negligible.
Impedance variations become less sensitive as a function of frequency as the ratio l /d decrease (diameter increase)
Broadband characteristics can be obtained by increasing the diameter of wire.
Figure. Center-fed cylindrical antenna configuration

6. Computational electromagnetic methodologies
Maxwell’s Equations are the basic set of equation describing the electromagnetic field.
Numerical methods
High Frequency
Differential equation
Integral equation
Field based
(Ray based)
Current based
Time-
domain
Frequency-
domain
Time-
domain
Frequency-
domain GO
Geometrical optics
TWTD
MoM
Method of Moment PO
Physical optics
FDFD
Finite-
Difference-
frequency-domain
FDTD
Finite-
Difference-
Time-domain
FEM
Finite- Element -Method)
GTD
Geometrical theory of diffraction
UTD
Uniform theory of diffraction
HFSS
CTS

7. [K]{u}={F} {u}= [𝐊] −1 {𝐅}
Computational electromagnetic methodologies
FEM : Finte-Element-Method
Many engineering phenomena can be expressed by “governing equations” and “boundary conditions”
Governing Equation
(Differential equation)
Boundary conditions
A set of simultaneous algebraic equations
[K]{u}={F} {u}= [𝐊] −1 {𝐅}
Property
Behavior
Action
FEM
It is very difficult to make the algebraic equations for the entire domain.
Discretizing solution regions into finite number of sub-regions or elements (in HFSS, used tetrahedron with triangular faces)
Deriving governing equations (elemental equation) for a typical element
Assembling of all elements (meshing) in the solution region to form matrix equation
Solving the system of equations obtained.

8. Computational electromagnetic methodologies
HFSS analysis design flowchart-adaptive mesh
HFSS sets up an initial mesh, solves the fields, and then re-meshes based on where the fields have a high concentration and/or gradient (mesh density).
Each re-meshing step is called an "adaptive pass".
Importantly, at each step, the scattering parameters are evaluated at each port, and compared to the previous step. The difference between the two is called "delta S (S parameter)’.

9. Computational electromagnetic methodologies
FDTD : Finite-Difference-Time-Domain
Represent the derivatives in Maxwell’s curl equations by finite differences.
Use the second-order accurate central difference formula.
A three-dimensional problem space is composed of cells

10. Computational electromagnetic methodologies
FDTD : Finite-Difference-Time-Domain
The Yee cell
direct solution of Maxwell’s Time-dependent curl equations
Partial differential equations will be approximated by replacing the time/space derivations with finite differences
Six(coupled) differential equation
: 𝐸 𝑥 , 𝐸 𝑦 , 𝐸 𝑧 , 𝐻 𝑥 , 𝐻 𝑦 , 𝐻 𝑧
Finite-Difference Equation for 𝐸 𝑥
2x2x2

11. Continuous system -> discrete system
Computational electromagnetic methodologies
FDTD : Finite-Difference-Time-Domain
Continuous system -> discrete system
Updating equations
Express the future components in terms of the past components.
Electric and magnetic fields are sampled at alternate times.

12. Comparison of FEM, FDTD
Both FEM and FDTD require that the objects being simulated are placed into a ‘box’ which truncates space and defines the simulation domain.
FEM allows designers to simulate most arbitrary 3D geometries however for geometrically complex and/or electrically large structures, the mesh can become very complex with many tetrahedral mesh cells.
Necessary for huge matrices to solve which can require very large amounts of computer memory
Frequency domain-> more appropriate for the analysis of ‘High Q’ circuits
ex) filters, cavities, resonators etc.
Most efficient solution to problems with large number of ports.
Ex) IC packages and multi-chip modules
Figure 1. Typical tetrahedral mesh used in FEM simulation
FDTD does not require a matrix solve and can often be addressed using small amount of computer memory .
Small number of ports but is electrically large, the most memory efficient.
Time domain -> useful for performing time domain reflectometry(TDR) analysis on connector interfaces & transitions (employed more in the high speed digital/signal integrity world)
Applications better suited to FDTD simulation include the likes of antenna placement on vehicles/aircraft and the analysis of antenna performance in the presence of detailed human body models.
Figure 2. Typical hexahedral mesh(Yee cell) used in FDTD simulation

13. Computational electromagnetic methodologies
GO (Geometric Optics) / GTD(Geometrical Theory of Diffraction) / UTD(Uniform Theory of Diffraction)
Ray tracing
The well-known two-ray model uses the fact that for most wireless propagation cases, two paths exist from transmitter to receiver: a direct path and a reflected path (bounce off the ground). Rx Tx
Figure. Example of ray tracing in a simple outdoor scenario
Intended for the consideration of electrically large dielectric structures in applications.
GO describes the direct propagation along a straight line path between Tx and Rx (direct ray), reflected rays, transmitted rays.
GO ignore the edge diffractions. As an extension of GO, GTD describes the propagation by diffraction from the edges between two surfaces in terms of diffracted rays.
UTD is a uniform version of the GTD. (GTD produce inaccurate results at the shadow boundaries)
These rays enable the EM waves to reach the shadowed regions of an environment which have otherwise have no access to the fields via direct paths (line of sight)

14. 4.8.2 Horizontal electric dipole
Perpendicular (horizontal or E) polarization
Parallel (vertical or H) polarization
Plane of incident : x-z plane
Figure 1 : Perpendicular (horizontal or E) polarized uniform plane wave incident at an oblique angle on an interface
Figure 2 : Parallel (vertical or H) polarized uniform plane wave incident at an oblique angle on an interface

15. 4.8.2 Horizontal electric dipole
Figure 4.25
z>0, air
z=0, ground
(a) Horizontal electric dipole above ground plane
(b) Far-field observations
(4-111)
(4-113)
(4-112),(4-112a)
Reflection coefficient
(4-128)
(4-129)

16. 4.8.2 Horizontal electric dipolex y
𝐸 𝑖
Figure 4.25 Horizontal electric dipole above ground plane

17. 4.8.2 Horizontal electric dipole
Grazing angles( θ i → 90°)
Figure 4.31 Elevation plane amplitude patterns of an infinitesimal horizontal dipole above a perfect electric conductor(σ =∞) and a flat earth (σ1 = 0.01 S/m, Ir1 = 5, f = 1 GHz)
The relative pattern of a horizontal dipole above a lossy surface is similar to that above a perfect electric conductor

18. Reflecting surface (seawater,..)
4.8.3 Earth Curvature
Antenna pattern measurement - Full scale model measurement
outdoor condition
Reflecting surface (seawater,..)
Unwanted signals : multipath
Desired signals :direct signals
Total measured signal in an outdoor system configuration is combination of direct signal and that due to multipath.
It is necessary to be able to subtract from the total response the contributions due to multipath.
Developing analytical models to predict the contributions due to multipath is necessary.
The curvature of the earth has tendency to spread out (weaken, diffuse, diverge) the reflected energy more than a corresponding flat surfaces.

19. 4.8.3 Earth Curvature Geometrical consideration (4-131)
Figure 4.33 (a) Reflection form a flat surface
Figure 4.33 (b) Reflection form a spherical surface

20. 4.8.3 Earth Curvature
Reflecting angle
height
The tangent at the point of reflection
distance
Principal radii of curvature of the reflected wave-front at the point of reflection
Figure 4.34 Geometry for reflections from a spherical surface
(4-132a)
(4-132)
(4-132b)
(4-132c)

21. 4.8.3 Earth Curvature A simplified form of the divergence factor
(4-133)
Assuming that the divergence of rays in the azimuthal plane is negligible
(4-134)
Figure 4.34 Geometry for reflections from a spherical surface
The divergence factor can be included in the formulation of the fields radiated by a vertical or a horizontal dipole, in the presence of the earth,
(4-135a)
(4-135b)

22. 4.8.3 Earth Curvature
The roughness of any scattering surfaces is not an intrinsic property of that surface but depends on the properties of a wave being transmitted.
Both the frequency and the local angle of incidence of the transmitted wave, determine how rough or smooth any surface appears to be.
frequency : The relation of the EM wave in terms of its wavelength λ to the statistical roughness parameter s is given by ks.(𝑘= 2𝜋 λ )
Angle of incidence of the transmitted wave
Ideal case(smooth surface/ PEC) : Fresnel reflectivity
Non-ideal case(rough surface) : Rayleigh and Frauenhofer criterion.
Figure. Rates of roughness components demonstrated on a (a)smooth (b)rough and (c) very rough surface
The backscattered EM wave on a surface consists of two components.
Coherent(reflected) component reacts as a specular reflection on a smooth surface.
Incoherent(scattered) component is a diffuse scatterer and distributes the scattering power in all directions.
As the surface becomes rougher, the coherent component becomes negligible and the incoherent component consist of only diffuse scattering.

23. 4.8.3 Earth Curvature
Figure. Diagram for determining the phase difference between two parallel waves scattered from different points on a rough surface
Phase difference ∆ϕ between rays scattered from separate points on the surface (1)
The Rayleigh criterion states that if the phase difference ∆ϕ between two reflected waves is less than π/2 radians,
than the surface may be considered as smooth, and is defined by (4-136)
Since the dividing line between a smooth and a rough surface is not that well defined, (4-136) should only be used
as a guideline.

24. 4.8.3 Earth Curvature
Vertical and horizontal polarization smooth surface reflection coefficients
(4-125)
(4-128)
The coherent contributions due to scattering by a surface with Gaussian rough surface statistics
Phase difference ∆ϕ
(4-137)
where
𝑅 𝑣,ℎ 𝑠 = reflection coefficient of a rough surface for either vertical or horizontal polarization
𝑅 𝑣,ℎ 0 = reflection coefficient of a smooth surface for either vertical (4-125) or horizontal (4-128) polarization
ℎ 0 2 = mean-square roughness height
Slightly rough surface : rms height << wavelength
Very rough surface : rms height >> wavelength

25. 4.8.3 Earth Curvature
Observation point ℎ 2 ′
Divergence factor approaches unity as the grazing angle becomes larger
Divergence factor is smaller than unity
Figure 4.34 Geometry for reflections from a spherical surface
Figure 4.35 Divergence factor for a 4/3 radius earth ( 𝑎 𝑒 =5,280 mi=8,497.3km)
as a function of grazing angle ψ
The determination of the reflection point(to find 𝑑 1 and 𝑑 2 ) from a knowledge of the heights of the source( ℎ 1 ′ ), observation points( ℎ 2 ′ ), and range d between them.
(4-138a)
(4-138b)
(4-138c)
(4-138d)
The cubic equation
(4-138)

26. 4.8.3 Earth Curvature
Height gain is defined as the ratio of the total field in the presence of the earth divided by the total field in the absence of the earth (received in free space)
Figure 4.36 Measured and calculated height gain over the ocean

27. 5.1 Introduction
Different forms of loop antenna

28. 5.1 Introduction
Loop antennas are usually classified into two categories. Electrically small and electrically large.
Circumference of electrically small loops : C < λ/10
Circumference of electrically large loops : C ∼ λ
HF (3-30MHz), VHF ( MHz), UHF (300-3,000MHz) bands.
Electrically small loops have small radiation resistances that are usually smaller than their loss resistances (very poor radiators).
They are employed for
transmission in radio communication (usually in the receiving mode) where antenna efficiency is not as important as signal-to-noise ratio.
Probes for field measurements as directional antenna for radiowave navigation
Way to increase the radiation resistance of the loop
Increasing its perimeter/ the number of turn
Insert a ferrite core of very high permeability within its circumference of perimeter
loop
ferrite core
Figure. Loopstick antenna(ferrite-rod antenna)

29. 5.1 Introduction
Similar to the field pattern of infinitesimal dipole with null perpendicular to the plane of the loop and its maximum along the plane of the loop
C increases
Maximum of the pattern : plane of the loop
axis of the loop

30. 5.1 Introduction
Electrically large loops are used primarily in directional arrays.
To achieve directional pattern characteristics, C≃λ.
The proper phasing between turns enhances the overall directional properties.
Figure. Yagi-Uda arrays
Maximum radiation is directed toward the axis of the loop
Figure amplitude power pattern for end-fire mode
Figure. Helical antenna
Figure. Quad arrays

31. 5.1 Introduction Plane of antenna
ground
Figure (a) Single element
Figure (b) Loop array
Mounting orientation of the loop will determine its radiation characteristics relative to the ground.
Figure. Three axis orthogonal loop array for receiver applications
Figure. Three axis symmetric loop array for receiver applications
Figure. Delta loop array

32. 5.2 Small circular loop The wire is assumed to be very thin (5-1)
Figure 5.2 (a) Geometry for circular loop

33. 5.2.1 Radiated field (5-2) (5-3) Cylindrical component (5-4)
Figure 5.2 (a) Geometry for circular loop
(5-2)
(5-3)
Cylindrical component
(5-4)
more convenient
Spherical unit vectors
(5-6)
(5-7)
Figure 5.2 (b) Geometry for far-field observation

34. 5.2.1 Radiated field (5-8) (5-9) (5-10)(5-11)
Figure 5.2 (b) Geometry for far-field observation
(5-2)
(5-12)
(5-13)
(5-14)

35. 5.2.1 Radiated field (5-14) Expanded in a Maclaurin series in a using
(5-15b)(5-15c)
(5-15d)
(5-16)
(5-16a)
(5-16b)
(5-17)

36. 5.2.1 Radiated field (5-17) (3-2a) (5-18a) (5-18b) (5-18c) (3-15)

37. 5.2.2 Small loop and infinitesimal magnetic dipole
Figure. Circular / square electric loop l
𝐼 𝑚
Magnetic dipole
Field produced by a small loop of electric current at large distances are the same as those produced by a linear magnetic dipole(bar magnet) of length l.
(4-10c)(4-8a)
(4-10a)
(4-10b)
(4-8b)
(5-20a)
(5-20b)
(5-20c)
(5-20d)
duality

38. 5.2.2 Small loop and infinitesimal magnetic dipole
(5-20a)-(5-20d) are compared with (5-18a)-(5-19b)
Figure 5.2 (a) Geometry for circular loop
(5-20a)
(5-20b)
(5-20c)
(5-20d)
(5-18a)
(5-18b)
(5-18c)(5-19a)
Magnetic dipole of magnetic moment 𝐼 𝑚 𝑙 is equivalent to a small electric loop of radius a and constant electric current 𝐼 0
(5-19b)

39. Thank you for attention.

40. Computational electromagnetic methodologies
PO (Physical Optics)