1. Antenna Theory
By CONSTANTINE A.BALANIS
Ch5.4.1~5.7.2
O Yeon Jeong

2. Contents 5. Loop Antennas 5.4.1 Arrays 5.4.2 Design procedure
Ground and earth curvature effects for circular loops
Polygonal loop antennas
5.6.1 Square loop
5.6.2 Triangular, Rectangular, and rhombic loops
Ferrite loop
5.7.1 Radiation Resistance
5.7.2 Ferrite-loaded receiving loop

3. 5.4.1 Arrays Helical antenna
A wire antenna, which is wound in the form of a helix.
In most cases the helix is used with a ground plane. The ground plane can take different forms.
(10-24) z
Figure 1.1 Helical antenna with ground plane
Ground plane
Special cases of Helical antenna
𝛼=0°→𝑆=0 : loop antenna
𝛼=90°→𝐷=0 : dipole antenna

4. 5.4.1 Arrays Mode of helical antenna A. Normal mode B. Axial mode
Overall length < λ
Omnidirectional pattern (similar to a dipole or a small loop).
Compact size, used as a monopole antennas for mobile cell or cordless telephone.
B. Axial mode
3/4 λ< C < 4/3 λ, C≃λ optimum (C : circumference of the loop)
End-fire mode (only one major lobe and the minor lobes -> directional)
General polarization of the antenna is elliptical (circular and linear polarizations can be achieved over different frequency range).
Figure 1.2 Three-dimensional normalized amplitude linear power patterns for normal and end-fire helical designs
(a) normal mode
(b) axial mode
(a) normal mode
(b) axial mode
Figure 1.3 Normal mode for helical antenna and its equivalent
N small loops
N short dipoles

5. 5.4.1 Arrays
Yagi-Uda array
Driven element : The RF power is fed into only one of the antennas.
Other elements(Parasitic elements) get their RF power from the driver through mutual impedance.
Parasitic elements : Elements are not electrically connected.
Reflector
Are made 5% longer than the driven element.
Reverses the direction of energy emitted from rear of antenna
Director
Are made 5% shorter than the driven element.
Reinforces and focuses energy form the front of the antenna
Most Yagi antennas have 1 reflector and 1-20 directors.
Reflector
Directors
Support boom
Driven element
Figure 2.1 Yagi-Uda array of dipole elements
Direction of maximum radiation
Figure 2.2the radiation pattern with and without the reflector

6. 5.4.1 Arrays
Yagi-Uda array
The greater the number of directors, the higher the gain and the narrow the beamwidth.
dipole
Gain 2.16 dBi
beamwidth 77.4 deg
Front/Side dB
2 element Yagi
Gain 5.37 dBi
beamwidth 73.1 deg
Front/Back 9.48 dB
3 element Yagi
Gain 7.94 dBi
beamwidth 53.2 deg
Front/Back 4.71 dB
4 element Yagi
Gain 8.77 dBi
beamwidth 40.7 deg
Front/Back 3.65 dB
5 element Yagi
Gain 8.6 dBi
beamwidth 37.7 deg
Front/Back 8.02 dB
6 element Yagi
Gain dBi
beamwidth 34.8 deg
Front/Back 5.88 dB
Figure 2.3 Polar pattern of Yagi-Udan array

7. 5.4.1 Arrays Yagi-Uda array Quad antenna
As the circumference of the loop increases, the radiation along its axis increases and reaches near maximum at about λ.
Thus loops can be used as the basic elements, instead of the linear dipoles
Figure 2.4 Yagi-Uda array of circular loops
Figure 2.5 Three-dimensional pattern for loop antenna of varying circumference
Quad antenna
Used for ham radio operatiors.
The overall perimeter of each loop ≃ λ
Figure 2.6 quad antenna
λ 4
Figure 2.7 Development of the (cubical) quad antenna
(a) Stacked dipoles
(b) Ends bent towards each other

8. 5.4.2 Design Procedure The design of small loop
(Uniform current distribution)
The design of large loop
(Non-uniform current distribution)
Radiation resistance
(5-24)
Directivity
(5-31)
Figure 3.1 Directivity of circular-loop antenna for 𝜃=0,𝜋 versus electrical size
Maximum effective area
(5-32)
(a) Resistance
(b) Reactance
Figure 3.2 Input impedance of circular-loop antennas
Resonance capacitance
(5-35)
Inductance
(5-37a)
(5-37b)

9. 5.4.2 Design Procedure Example 5.4
Design a resonant loop antenna to operate at 100MHz so that the pattern maximum is along the axis of the loop.
Determine the radius of the loop and that of the wire (in meter), the axial directivity (in dB), and the parallel lumped element.
Solution
Self resonant loop, there is no need for a resonant capacitor.
Choosing an Ω = 12, C≃1.125λ
Free-space wavelength λ=c∙𝑓=3 meters
Circumference ≃1.125(3)=3.375 meter
Pattern maximum along the axis of the loop
Current distribution will be non-uniform.
Figure 3.3 Three-dimensional pattern for loop antenna of varying circumference
Figure 3.2 (b) Input reactance of circular-loop antenna

10. 5.4.2 Design Procedure Example 5.4
Design a resonant loop antenna to operate at 100MHz so that the pattern maximum is along the axis of the loop.
Determine the radius of the loop and that of the wire (in meter), the axial directivity (in dB), and the parallel lumped element.
Solution
Using Figure 3.1, the axial directivity for this design is approximately 3.6 dB. Using Figure 3.2 (a),
The input impedance is approximately 𝑍 𝑖𝑛 ≈125 ohms
Figure 3.1 Directivity of circular-loop antenna for 𝜃=0,𝜋 versus electrical size
Figure 3.2 (a) Input resistance of circular-loop antennas

11. 5.5 Ground and earth curvature effects for circular loops
The reflections are taken into account by introducing appropriate image source.
The effect of the ground curvature is taken into account by introducing divergence factor.
Figure 4.2 Geometry for reflections from a spherical surface
Figure 4.1 Vertical electric dipole above an infinite, flat, perfect electric conductor

12. 5.5 Ground and earth curvature effects for circular loops
The directivity variations are not strongly influenced by the radius of the wire.
The variation of impedance do show a dependence on the radius of the wire of the loop for 10<Ω<20.
When a resonant loop is close to the interface, the changes in input admittance as a function of the antenna height and electrical properties of the lossy medium were very pronounced.
ground
Figure 4.3 Directivity of circular-loop antenna, C = ka = 1,
for θ = 0 versus distance from reflector h / λ.
Theoretical curve is for infinite planar reflector.
Figure 4.4 Input impedance of circular-loop antenna C = ka = 1 versus distance from reflector h / λ. Theoretical curves are for infinite planar reflector; measured points are for square reflector

13. 5.6 Polygonal loop antennas
Usually, the circular loop has been used in the UHF range because of its high directivity while triangular and square loops have been applied in the HF and UHF bands because of advantages in their mechanical construction.
Broadband impedance characteristics can be obtained from the different polygonal loops.
Figure 5.1 Triangular loop antenna
Figure 5.3 Square loop antenna
Figure 5.4 rhombic loop antenna
Figure 5.2 Rectangular loop antenna

14. 5.6.1 Square loop (4-26a) (4-26b) (4-26c)
By assuming that each of small loop sides is a small linear dipole of constant current 𝐼 0 and length a.
(5-68)
Figure 6.1 Square loop geometry for far-field observations on the y-z plane
(5-68a)
(5-68b)
(5-69)

15. 5.6.1 Square loop (5-69) For small values of a (a < λ / 50) (5-27b)
(5-70)
(5-71)
Figure 6.1 Square loop geometry for far-field observations on the y-z plane
Figure 6.2 Circular loop geometrical arrangement for far-field observations

16. 5.6.2 Triangular, rectangular, and rhombic loops
Parameters
The radius of wire : b
The angle of the top corner of the isosceles triangle for the triangular and rhombic loop : 𝛽
The relative side dimensions of the rectangular loop : 𝛾 =W/ H
Figure 7.3 Typical configurations of polygonal loop antennas
Feed at the top corner of the isosceles triangle
Feed at its terminals at the base
Feed at its terminals at one of its corners
Feed at the center of one of its side

17. 5.6.2 Triangular, rectangular, and rhombic loops
Impedance characteristics
Restricted to a wire radius of a/L = (rhombic loop : a/L = ) -> very thin wire.
Frequency interval between adjacent dots on each locus is 𝐿 λ =0.2.
The input resistance value near the resonance frequency changes depending on the change of loop shape.
Input impedance value to match the characteristic impedance of a given transmission line can be obtained by changing the shape of the loop.
Impedance variation with respect to wire radius is very similar to that of dipole antennas.
Figure 7.4 (a) Impedance characteristics of top-driven triangular loop

18. 5.6.2 Triangular, rectangular, and rhombic loops
Small radius of the locus
-> slight variation of the input impedance
-> smaller variation
-> broad bandwidth characteristic
Figure 7.5 Reflection coefficient in dB versus frequency
Narrow bandwidth characteristic
Broad bandwidth characteristic

19. 5.6.2 Triangular, rectangular, and rhombic loops
Impedance characteristics
(b)
(a)
Triangular loop
The radius of the locus for the top-driven loop (𝛽=60°, marked red line) is much smaller than the others.
It has the broadest impedance bandwidth compared with other triangular shapes, and with the same shape but with different feed-points.
(d)
(c)
Rectangular loop
- For a rectangle of 𝛾=0.5 (marked blue line), broadest impedance bandwidth
Figure 7.4 Impedance characteristics
Top-driven triangular loop
Rectangular loop
Base-driven triangular loop
Rhombic loop

20. 5.6.2 Triangular, rectangular, and rhombic loops
Current distribution
Resonances are of two type
(True) resonances :
the reactance changes from negative to positive (capacitive to inductive)
associated with frequency of minimum impedance -> current flows freely
Anti-resonances :
the reactance changes from positive to negative (inductive to capacitive) ->reverse occurs
associated with frequency of maximum impedance ->little current flows
All true harmonic resonances occur near integer values of 𝑘 𝑏 , since this is where the wave is reinforce.
Figure 7.1 Geometry of the loop
Maximum impedance at anti-resonance
Minimum impedance at anti-resonance
Impedance, ohms
Figure 7.2 Plot of the impedance for a loop
size Ω = 12, b/a≃64

21. 5.6.2 Triangular, rectangular, and rhombic loops
Current distribution
Figure 7.5 current distribution of polygonal loops at anti-resonant wavelength
When a current null in the standing wave distribution hits a sharp corner of the loop at its anti-resonant frequency,
the loop has a current pattern of very low standing wave ratio and, in turn, broad-band impedance characteristics.

22. 5.6.2 Triangular, rectangular, and rhombic loops
Current distribution
Low frequency near the resonance, there is much less difference between the gains of a top-driven triangular loop and base-driven loop antennas. (Similar pattern : 𝐿 λ =1.0 : , 𝐿 λ = )
Higher frequency, the base-driven loop has a greater gain than the top-driven loop antenna.
Figure 7.6 Radiation pattern
(a) Top-driven triangular loop
(b) Base-driven triangular loops
(c) Rectangular loops

23. 5.7 Ferrite loop
The radiation resistance of loop antenna is comparable with its loss resistance.
Therefore, the radiation efficiency of loop antenna is quite low.
To improve, radiation efficiency of a loop antenna,
Increasing the circumference of loop
Employing ferrite rods
Figure 8.1 Radiation resistance for a constant current circular-loop antenna
Figure 8.2 ferrite rod antenna

24. Magnetic flux is concentrated inside the ferrite rod
5.7.1 Radiation Resistance
Ferrite materials are known for high permeability and high resistivity.
The ferrite core has a tendency to increase the magnetic flux (field), the open-circuit voltage, and in turn the radiation resistance of the loop.
(5-72)
where
𝑅 𝑓 = radiation resistance of ferrite loop
𝑅 𝑟 = radiation resistance of air core loop
𝜇 𝑐𝑒 = effective permeability of ferrite core
𝜇 0 = permeability of free-space
𝜇 𝑐𝑒𝑟 = relative effective permeability of ferrite core
Magnetic flux is concentrated inside the ferrite rod
(b) Concentration of magnetic flux
(a) construction
(c) Radiation pattern
Figure 8.3 Ferrite rod antenna

25. 5.7.1 Radiation Resistance
The relative effective permeability of the ferrite core 𝜇 𝑐𝑒𝑟 is related to the relative intrinsic permeability of the unbounded ferrite material 𝜇 𝑓𝑟 = 𝜇 𝑓 / 𝜇 0
(5-75)
where D is the demagnetization factor which has been found experimentally for different core geometries
(5-75a)
The radiation resistance of loop
(5-24)
(5-24a)
For a small ferrite loop
(5-73)
(5-74)
Figure 8.4 Demagnetization factor as a function of core length/diameter ratio

26. 5.7.2 Ferrite-loaded receiving loop
The combination of a low resistance with a large inductance can make it difficult to match the antenna as a source or load.
To deal with this problem, a capacitor is used to turn the loop into a resonant circuit / antenna.
Figure 8.5 (a) Tuned ferrite rod antenna equivalent circuit
𝐿 𝑓
𝑅 𝑟 𝑅
𝑅 𝑓 𝑐
Figure 8.5 (b) Tuned ferrite rod antenna equivalent circuit
𝑄 2 𝑅 𝑟
𝑄 2 𝑅
𝑄 2 𝑅 𝑓
By using a suitable parallel capacitance (C), we can convert the antenna’s terminal impedance into a pure resistance whose magnitude is 𝑄 2 larger than the actual loop resistance at resonant frequency.
Resonant frequency
Quality factor

27. Thank you for attention.