Antenna is a radiator and sensor of EM waves. It is also a transducer and impedance matching device. It can be designed to direct EM energy in desired directions and suppress it in unwanted directions.
The main aim of this chapter is to provide the fundamentals of antennas. They include:
The general Maxwell’s equations and retarded potentials are repeated here for the sake of continuity and clarity.
∇ × H = Ḋ + J
∇ × E = − Ḃ
∇.D = ρυ
∇.B = 0
where |
H = magnetic field, (A/m) |
|
D = displacement electric flux density (c/m2) |
|
displacement electric current density (A/m2) |
|
J = conduction current density (A/m 2) |
|
E = electric field (V/m) |
|
B = magnetic flux density (wb/m2) |
|
magnetic current density (V/m2) |
From these equations, it is possible to obtain general solutions for E and H fields in terms of potentials and also expressions relating potentials and their sources.
By definition, vector magnetic potential, A is given by
|
∇ × A = B = μH |
or, |
∇ × Ȧ = μḢ |
But |
μḢ = −∇ × E |
|
∇ × Ȧ = −∇ × E |
or, |
∇ × (E + Ȧ) = 0 |
This is true only if (E + Ȧ) represents the gradient of a scalar. Hence,
And we know
or,
Therefore, the general expressions for E and H are given by
Expressions Relating Potentials and Their Sources
Consider |
∇ × A = B |
|
∇ × ∇ × A = μ ∇ × H |
or, |
∇ × ∇ × A = μ ∈ Ė + μJ |
or, |
∇ ∇.A − ∇2 A = μ ∈ [−∇ − Ȧ] + μJ |
|
= −μ ∈ ∇ − μ∈ Ȧ + μJ |
Curl of A is known and its divergence is not specified.
Helmholtz Theorem
This states that any vector like A has unique meaning only if its curl and divergence are specified.
Hence, the divergence of A is
This is known as Lorentz gauge condition.
Using Lorentz gauge condition, we get
or, ∇2A − μ ∈ Ȧ = − μJ
Now consider ∇.D = ρυ
or, ∇.E = ρυ/∈
|
∇.(−∇V − Ȧ) = ρυ/∈ |
or, |
∇2 V + ∇.Ȧ = − ρυ/∈ |
|
∇2 V − μ ∈ = −ρυ/∈ |
The final expressions are:
For sinusoidal fields, these equations become
The potentials for static fields are
But for time varying fields, they can be written as,
The potentials are usually established due to time varying field only after some amount of propagation time. This propagation time depends on the distance between the point of the potentials from their sources and velocity of propagation of EM fields. As a result, the potentials are retarded by a time, r/ ʋ0. These potentials are known as retarded potentials, and they are expressed as
It is well known that a uniform plane wave propagating in rdirection has a phase variation represented by e−jβr. Including this phase factor, the above potentials are given by
Antenna or Aerial means the same.
An antenna is defined in the following ways:
These properties can be proved using reciprocity theorem.
Z a is a complex quantity and it is written as
Here, the reactive part Xa results from fields surrounding the antenna. The resistive part, Ra is given by
Here Rl represents losses in the antenna. Rr is called radiation resistance.
or,
That is, is called field strength pattern.
That is, is called Power pattern.
More generally an antenna radiation pattern is a three dimensional variation of the radiation field.
Leff of transmitting antenna It is that length of an equivalent linear antenna that has a constant current along its length and which radiates the same field strength as the actual antenna.
Refer Fig. 8.1
.Fig. 8.1 Definition of effective length of transmitting antenna
Leff of transmitting antenna is defined mathematically as
Leff of receiving antenna It is defined as the ratio of the open circuit voltage developed at the terminals of the antenna under the received field strength, E, that is,
Effective length of an antenna is always less than the actual length.
Here |
η0 = intrinsic impedance of the medium, (Ω) |
r= radius of the sphere, (m) |
|
P= power radiated-instantaneous | |
E= electric field strength, (V/m) |
|
RI= RI(θ, ϕ) is a function of θ and ϕ |
or,
where wt = wr + wl ,
or,
where |
WR = received power (watt) |
|
P= power flow per square metre (watts/m2) for the incident wave |
Fig. 8.2 Antenna equivalent circuit
The main difference between the antenna equivalent circuit and an RLCcircuit is that Ra , La and Ca vary with frequency. As a result, the antenna conductance peak appears not at resonant frequency but at a frequency slightly away from fr (Fig. 8.3).
Fig. 8.3 Antenna conductance and susceptance variation
The basic antenna elements are:
When a transmitting antenna is excited with an alternating voltage, the initial motion of a wave which is propagated through space is started by the balanced motion of charges in the antenna. The transmitting antenna has characteristics similar to those of a resonant circuit.
When energy is supplied to it, resonant oscillations occur in the antenna and violent variations in charge form an electric vector. The same violent motions of charges create a magnetic field about the antenna in the same manner as a magnetic field expands and collapses about a resonant circuit tank coil.
If energy is continuously applied to the antenna, energy moves away from the antenna into space in the form of EM waves. It may be noted that it is not the original antenna charges themselves that move through space but rather the motion they create. The charges around the antenna are set in motion first and they, in turn, set other charges further separated from the antenna into motion. This disturbance fans out from the antenna into space.
Wave motion of charges forms an electric field and a magnetic field which result from the motion of charges. These electric and magnetic fields are perpendicular to each other. The motion of these fields in the form EM waves has no boundaries and expand spherically. The EM energy density decreases with distance as they propagate.
The concept of an alternating current element, Idlcos ωtis of theoretical interest. But the theory developed for this can be extended to practical antennas. To derive radiation fields of antenna elements including current element, the concept of retarded vector magnetic potential is very useful.
Derivation of radiation fields consists of the following steps:
Write expression for retarded vector magnetic potential.
Write expressions for the components of A in Cartesian coordinates.
Express A in the components of the spherical coordinate system.
Obtain the components of H from μ H = ∇× A.
Obtain the component of E from (as J = 0 for space).
Consider an alternating current element at the origin of a spherical coordinate system (Fig. 8.4).
Fig. 8.4 Alternating current element at the origin
The vector magnetic potential, A (r, t) is given by
As the element is z-directed, A is also z-directed.
The volume integral in the above equation can be simplified by taking integration over the cross-sectional area of the element and an integration along its length. We know,
and
This means A has only z-component and Ax = 0, Ay = 0. Changing Cartesian components into spherical coordinate components, we have
But we know
and
As Aϕ = 0 and Aθ ≠ f(ϕ), Hr = 0.
Similarly,
Here
Substituting the expressions of Aθ , Ar and simplifying, we get
From Maxwell’s first equation, we have
From this, we get
[as H = Hϕaϕ]
and
where
The total field components of an alternating current element are
The derivation of the expression for radiated power consists of the following steps:
Poynting vector is
P = E ×H xs watts/m2
P θ =−E r H ϕ
But
Using these identities, Pθ becomes
Pθ represents the instantaneous power flow in θ-direction. But the average value of cos 2ωtd or sin 2ωtd over a cycle is zero. Hence (Pθ) av = 0 at any value of r. This means power in θ-direction surges back and forth.
Similarly, Pr = Eθ Hϕ
Using the expressions of Eθ and Hϕ , we get
It is obvious that the average value of Pr is
or,
where
The total power radiated
Pr(av) is independent of ϕ and hence the elemental area ds on the spherical shell is
Now PT becomes
But
Here Iis the peak value of current.
As
Equation for PT becomes
or,
This is in the form of P= I2R. Hence the coefficient of has the dimensions of resistance and it is called Radiation Resistance.
Radiation Resistance of a Hertzian dipole
The field components of current elements are:
Hϕ field consists of terms and . term dominates over term at points close to the current element. When ris small, term is called Induction Field.
On the other hand, term dominates over term when ris large. This term is called Radiation Field or distant field or far-field.
The expression for Eθ consists of three terms, , , and and the expression for Er consists of and terms. The term is called Electrostatic Field.
In short,
term in E and H fields is called Radiation Field
term is called Induction Field
term is called Electrostatic Field
If the induction and radiation fields have equal amplitudes, then from the expression of Hϕ , we have
or,
At a distance of induction and radiation fields have equal amplitudes.
Hertzian dipole is defined as an infinitesimal current element Idlwhich does not exist in real life.
Hertzian dipole is a short linear antenna which, when radiating, is assumed to carry constant current along its length.
As Hertzian dipole and alternating current elements are virtually the same, the radiated power and radiation resistance are given by
The possible current distributions are:
1. Constant current along its length—valid in Hertzian dipole.
2. Triangular current distribution (Fig. 8.5).
Fig. 8.5 Short radiating elements
For triangular current distributions,
3. Sinusoidal current distribution (Fig. 8.6).
Fig. 8.6 Sinusoidal current distribution in dipole and monopole
4. Exact current distribution—This can be determined using the method of moment technique. The method is briefly presented in Chapter 9.
Radiated power by half wave dipole,
Radiation resistance of half wave dipole, Rr = 73Ω
Proof The proof consists of the following steps:
The sinusoidal current distribution is represented by Fig. 8.7.
Fig. 8.7 Sinusoidal current distribution
I= Im sin β (H− Z) for z > 0
= Im sin β (H+ Z) for z < 0
Here Im = current maximum
The vector potential at a point, Pdue to the current element I dzis given by
Here d is the distance from the current element to the point P. Let rbe the distance of Pfrom the origin. The total vector potential at Pdue to all current elements is given by
It is of interest here to consider radiation fields. din the denominator can be approximated to r. But in the numerator, dis in the phase term and it is given by
d = r − z cos θ
Now Az becomes
For a half wave dipole
But sinβ (H+ Z) = sinβ cosβz + cos β H sin βz
sinβ (H − Z) = sinβ cosβz − cos β H sinβz
As
sinβ (H+ Z) = sinβ (H− Z) = cosβz
Using these values, Az becomes
But
But we have
Hence
We also know that
The magnitude of E for the radiation field is
Eθ and Hϕ are in time phase. Hence the maximum value of Poynting vector is
The average value of Poynting vector is one half of the peak value.
or,
Therefore total power radiated through a spherical surface half wave dipole is
The numerical evaluation of the integral by Simpson’s or the Trapezoidal rule gives a value of 1.218.
As PT becomes
The coefficient of is nothing but radiation resistance. That is,
Radiated power of quarter wave monopole, , watts.
Radiation resistance, Rr = 36.5Ω
Proof Consider Fig. 8.8in which a monopole with current distribution is shown.
Fig. 8.8 Monopole
Obtain Pav exactly as described in half wave dipole, that is,
As the monopole is fed by a perfectly conducting plane at one end, it radiates only through a hemispherical surface. Therefore, the total radiated power is
Numerical evaluation of the integral by Simpson’s or the Trapezoidal rule gives a value of 0.609.
As
The Radiation resistance,
Electric field as a function θ in free space for a dipole of length 2His given by
The amplitude of Eθ is
The normalised (Eθ) is
The radiation patterns are the variation of ⎢Eθ ⎢n with θ. These patterns for different lengths of dipole are shown in Fig. 8.9.
Fig. 8.9 Radiation patterns of dipoles
Problem 8.1 Find the radiation resistance of a Hertzian dipole of length
Solution The radiation resistance of a Hertzian dipole of length dlis
Problem 8.2 Find the directivity of a current element, Idl.
Soultion The amplitude of electric far-field of a current element is
But
or,
Maximum radiation occurs at θ = π/2
The radiated power of the current element is
If wr is assumed to be 1 watt, then
The maximum radiation intensity is given by
The maximum directive gain, gd (max)
The directivity of current element
Problem 8.3 Find the directivity of a half wave dipole.
Soultion For a half wave dipole,
But Wr = 73I2 = watts
For Wr = 1 w,
Problem 8.4 An antenna whose radiation resistance is 300Ω operates at a frequency of 1 GHz and with a current of 3 amperes. Find the radiated power.
Soultion Radiated power,
Problem 8.5 What is the effective area of a half wave dipole operating at 500 MHz?
Solution The effective area of an antenna is
As
Directivity of half wave dipole is
Problem 8.6 Find the effective area of a Hertzian dipole operating at 100 MHz.
Solution As f= 100 MHz
Directivity of the Hertzian dipole,
Problem 8.7 An EM wave of 1 GHz is radiated by an antenna to cover a distance of 100 km. Determine the time taken by the wave to travel the above distance.
Solution The time taken by the EM wave is
where
Problem 8.8 The directivity of an antenna is 30 and it operates at a frequency of 100 MHz. Find its maximum effective aperture.
Solution D= 30
f= 100 MHz
Maximum effective aperture
Radiation intensity, watts/unit solid angle.
Directive gain,
Directivity, D= (gd) max .
Power gain,
Antenna efficiency,
Effective area,
Far-field is represented by field term.
Induction field is represented by field term.
Radiation resistance of a Hertzian dipole is
Electrostatic field is represented by term.
The far-field and induction field have equal magnitudes at
Radiation resistance of half wave dipole is 73Ω.
Radiation resistance of quarter wave monopole is 36.5Ω.
Horizontal pattern of vertical dipole is a circle.
Radiated power flow of a vertical dipole is in the radial direction.
1. An antenna is a transducer. |
(Yes/No) |
2. An antenna is a sensor of EM waves. |
(Yes/No) |
3. An antenna acts as an impedance matching device. |
(Yes/No) |
4. Effective length of a wire antenna is always greater than the actual length. |
(Yes/No) |
5. Directive gain = Power gain for an antenna. |
(Yes/No) |
6. The radiation fields are nothing but far-fields. |
(Yes/No) |
7. The radiation pattern of vertical and horizontal dipoles are identical. |
(Yes/No) |
8. The patterns of half wave dipole and quarter wave monopole are identical. |
(Yes/No) |
9. The radiated fields of z-directed half wave dipole consists of Eθ , Er , Hr , Hθ , terms. |
(Yes/No) |
10. The radiated fields of z-directed dipole consists of only Eθ , Er and Hϕ |
(Yes/No) |
11. Effective area of antenna is a function frequency. |
(Yes/No) |
12. Electric and magnetic dipoles have the same physical structure. |
(Yes/No) |
13. Magnetic dipole is a small current loop of wire. |
(Yes/No) |
14. Input and radiation resistances are the same. |
(Yes/No) |
15. The radiation resistance of a current element depends on frequency. |
(Yes/No) |
16. The differential current element is nothing but Hertzian dipole. |
(Yes/No) |
17. Delayed and retarded potentials mean the same. |
(Yes/No) |
18. Jdʋ = IdL |
(Yes/No) |
19. Radiated power of a current element depends on frequency. |
(Yes/No) |
20. The units of scalar and vector magnetic potentials are the same. |
(Yes/No) |
21. Electrostatic field contributes to the radiated power. |
(Yes/No) |
(Yes/No) |
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23. Electrostatic field does not contribute to radiation power. |
(Yes/No) |
24. Array increases the directivity. |
(Yes/No) |
25. Beam width is decreased by array. |
(Yes/No) |
26. If the number of elements are increased, the beam width is reduced. |
(Yes/No) |
27. Dipole is an omnidirectional antenna. |
(Yes/No) |
28. Isotropic and omnidirectional antennas mean the same. |
(Yes/No) |
29. Dipole and monopole mean the same except in length. |
(Yes/No) |
30. Power gain and directive gain are the same. |
(Yes/No) |
31. Radiation resistance of half-wave dipole is more than that of quarter monopole. |
(Yes/No) |
32. Power gain and efficiency of antennas are the same. |
(Yes/No) |
33. Effective area of receiving antenna depends on frequency. |
(Yes/No) |
34. The effective area of an antenna is independent of the length of the antenna. |
(Yes/No) |
35. The units of radiation intensity are _____ . |
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36. Directivity is _____ . |
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37. Efficiency of an antenna is _____ . |
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38. Efficiency of an antenna in terms of directive and power gains is _____ . |
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39. Effective area is _____ . |
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40. The far-field is indicated by the presence of _____ . |
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41. The induction field is indicated by the presence of _____ . |
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42. The electrostatic field is indicated by the presence of _____ . |
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43. The radiation resistance of an isolated half wave dipole is _____ . |
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44. The radiation resistance of a quarter wave monopole is _____ . |
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45. The current distribution in a half wave dipole is _____ . |
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46. The current distribution in alternating current element is _____ . |
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47. The current distribution in short dipoles is _____ . |
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49. The directivity of half wave dipole is _____ . |
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50. If a current element is x-directed, vector magnetic potential is _____ . |
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51. Radiation resistance of a short monopole is _____ . |
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52. Radiation resistance of a short dipole is _____ . |
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53. At LF and VLF, polarisation often used is _____ . |
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54. dBi means _____ . |
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55. dBm means power gain in dB _____ . |
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56. Antenna used in mobile communications is _____ . |
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57. If a current element is z-directed, vector magnetic potential is _____ . |
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58. If vector magnetic potential has only Az , Eϕ is _____ . |
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59. Radiation resistance of current element is _____ . |
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60. Radiation resistance of quarter wave monopole is _____ . |
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61. Directional pattern of a short dipole in the horizontal plane is a _____ . |
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62. Directional pattern of a horizontal halfwave centre-fed dipole is _____ . |
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63. Effective length of a dipole is always _____ than the actual length. |
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64. The directivity in dB of half wave dipole is _____ . |
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65. The directivity in dB of current element is _____ . |
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66. Effective area of a Hertzian dipole operating at 100 MHz is _____ . |
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67. The radiation pattern of a horizontal dipole is of _____ . |
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68. The radiation pattern of vertical dipole is of _____ . |
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69. Vector magnetic potential has the unit of _____ . |
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70. Retarded magnetic potential has the unit of _____ . |
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71. Radiated power is contributed by _____ only. |
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72. The radiation resistance of quarter wave dipole is _____ . |
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74. The total resistance of an antenna is _____ . |
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75. Power gain of an antenna is _____ . |
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76. The antenna of za impedance radiates maximum power when the transmitting line feeding the antenna has an impedance of _____ . |
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77. Directive gain of Hertzian dipole is _____ . |
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78. If the signal level is 1 mW, power gain is _____.
79. Whip antenna has a physical length of _____.
80. For a 300 Ω antenna operating with 5 A of current, the radiated power is _____.
1. Yes |
2. Yes |
3. Yes |
4. No |
5. No |
6. Yes |
7. No |
8. No |
9. No |
10. Yes |
11. Yes |
12. No |
13. Yes |
14. No |
15. Yes |
16. Yes |
17. Yes |
18. Yes |
19. Yes |
20. No |
21. No |
22. Yes |
23. Yes |
24. Yes |
25. Yes |
26. Yes |
27. Yes |
28. No |
29. No |
30. No |
31. Yes |
32. No |
33. Yes |
34. No |
|
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35. Watts/unit solid angle
36. Maximum directive gain
37.
38. gp /gd
39.
40. term
41. term
42. term
43. 73Ω
44. 36.5Ω
45. Sinusoidal
46. Constant
47. Triangular
48. 1.5
49. 1.64
50. x-directed
51.
52.
53. Vertical
54. Power gain of the antenna in dB relative to isotropic antenna
55. Compared to 1 mW
56. Whip antenna
57. z-directed
58. Zero
59.
60. 36.5Ω
61. Circle
62. Figure of eight
63. Less
64. 2.15
65. 1.64
66. 1.07 m2
67. Figure of eight shape
68. Dumbell shape
69. Wb/m
70. Wb/m
71. Far-field only
72. 36.5Ω
73. 73Ω
74. Rr + Rl
75. The product of efficiency and gain
76. Za *
77. 1.5
78. 0 dBm
79. λ/4
80. (a)
1. (c) |
2. (c) |
3. (b) |
4. (a) |
5. (a) |
6. (b) |
7. (a) |
8. (a) |
9. (c) |
10. (a) |
11. (b) |
12. (b) |
13. (b) |
14. (a) |
15. (a) |
16. (b) |
17. (a) |
18. (a) |
19. (a) |
20. (b) |