Radiation Pattern

Consider the fields of an infinitesimal Hertzian electric dipole of length dl kept at the origin of the coordinate system. Let I0 be the z-directed current in the Hertzian dipole radiating into free space. In spherical coordi-nates, the expressions for the E and H fields are given by Eqns (1.59)–

(1.61) and Eqns (1.64)–(1.66), respectively and are reproduced here for convenience

E_r = ηI₀dl cos θ 2π e^−jkr

1 r² + 1

jkr³



(2.1) Eθ= jηkI₀dl sin θ

4π e^−jkr

1 r + 1

jkr² − 1 k²r³



(2.2)

E_φ= 0 (2.3)

Hr = 0 (2.4)

H_θ= 0 (2.5)

H_φ= jkI₀dl sin θ 4π e^−jkr

1 r + 1

jkr²



(2.6)

where the impedance of the medium, η, is given by

η = k ω =

μ

 Ω (2.7)

In Eqn (2.7) μ and  are the permeability and permittivity of the medium, respectively. For free space, η =μ0/0 = 376.73 Ω. This is generally approximated to 377 Ω or 120π Ω.

The above field expressions are valid everywhere except on the dipole itself. Since the 1/r² and 1/r³ terms decay much faster than the 1/r term, at large distances from the dipole (r λ) we can neglect these terms and simplify the field expression by taking only the 1/r terms. This simplified expression is known as the far-field expression. Further, the Poynting vector corresponding to the 1/r² and 1/r³terms is reactive and that due to the 1/r term is real. Therefore, the 1/r terms are taken as the radiated fields. The region far away from the antenna is known as far-field or Fraunhofer region.

(The region close to the antenna is known Fresnel or radiating near-field region.) In the far-field region, only E_θ and H_φ exist and form a spherical wavefront emanating from the antenna. Thus, the far-field expressions for

E and H reduce to

E = a_θE_θ = a_θjηkI₀dl sin θ 4π

e^−jkr

r (2.8)

H = a_φH_φ= a_φjkI₀dl sin θ 4π

e^−jkr

r (2.9)

In the far-field, E_θand H_φare perpendicular to each other and transverse to the direction of propagation. The ratio of the two field components is the same as the intrinsic impedance, η, of the medium

E_θ

H_φ = η (2.10)

Thus, the radiated EM wave satisfies all the properties of a transverse elec-tromagnetic (TEM) wave.

The time-averaged power density vector of the wave is given by

S = 1

2Re(E× H^∗) = ar1

2E_θH_φ^∗ = ar 1

2η|Eθ|² W/m² (2.11) where H^∗ indicates the complex conjugate of H. Substituting the value of E_θ from Eqn (2.8), the time-averaged power density or the Poynting vector for a Hertzian dipole reduces to

S(r, θ, φ) = ar1

2η^_kI0dl 4π

²sin²θ

r² (2.12)

This shows that in the far-field of the antenna, the power flows radially outward from the antenna, but the power density is not the same in all directions.

EXAMPLE2.1

A Hertzian dipole of length dl = 0.5 m is radiating into free space. If the dipole current is 4 A and the frequency is 10 MHz, calculate the highest power density at a distance of 2 km from the antenna.

Solution: Since the antenna is radiating into free space, we can choose the coordinate system such that the dipole is oriented along the z-axis. The power density is given by Eqn (2.12) and has a maximum along θ = π/2.

Therefore, the maximum power density of a Hertzian dipole is given by

S = 1

2η^_kI0dl 4π

² 1

r² W/m² The propagation constant, k, is given by

k = ω√ μ = ω

c = 2π× 10 × 10⁶ 3× 10⁸ = 2π

30 rad/m Substituting the values of I₀ = 4, dl = 0.5, k, and r = 2000

S = 1

2 × 120π^_2π

30 ×4× 0.5 4π

² 1

2000² = 5.24× 10⁻⁸ W/m²

Consider an antenna kept at the origin of a spherical coordinate system and let S(r, θ, φ) be the average radial power density at a distance r from the antenna along the direction (θ, φ). Let dA be an elemental area on the surface of the sphere of radius r that subtends a solid angle dΩ at the centre of the sphere (which is also the origin of the coordinate system). The elemental area dA is given by

dA = r²dΩ m² (2.13)

In spherical coordinates dA = r²sin θdθdφ and hence the elemental solid angle dΩ (unit: steradian, sr) is given by

dΩ = sin θdθdφ sr (2.14)

EXAMPLE2.2

Show that there are 4π steradians in a sphere.

Solution: The total solid angle in a sphere is Ω =

 _π

θ=0

 _π

φ=0dΩ Substituting the expression for dΩ from Eqn (2.24)

Ω =

 _π

θ=0

 _π

φ=0

sin θdθdφ

Integrating with respect to φ and then with respect to θ Ω = 2π

 _π

θ=0sin θdθ = 2π[− cos θ]^π₀ = 4π sr

The power crossing the area dA is given by S(r, θ, φ)dA and the power crossing per unit solid angle is defined as the radiation intensity, U (θ, φ), and is given by

U (θ, φ) = S(r, θ, φ)dA

dΩ = r²S(r, θ, φ) W/sr (2.15) The power pattern or the radiation intensity, U (θ, φ), of an antenna is the angular distribution of the power per unit solid angle. This can be obtained by multiplying the Poynting vector by r². Thus, the radiation intensity pattern of a Hertzian dipole is

r²S(r, θ, φ) = U (θ, φ) = 1

2η^_kI0dl 4π

²sin²θ W/m² (2.16)

The normalized power pattern, P_n, is obtained by normalizing the radiation intensity, U , or the time-averaged Poynting vector, S, with respect to their maximum values.

Pn(θ, φ) = U (θ, φ)

Umax = S(r, θ, φ)

Smax(r) (2.17)

The normalized power is a dimensionless quantity and it is expressed in decibels as

P_ndB(θ, φ) = 10 log₁₀[P_n(θ, φ)] (2.18) For a Hertzian dipole it is given by

PndB(θ, φ) = 10 log₁₀(sin²θ) (2.19) A plot of the far-field electric or magnetic field intensity as a function of the direction at a constant distance from the antenna is known as the electric field pattern or the magnetic field pattern, respectively. Dividing the field quantities by their respective maximum values we get the normalized field patterns. For a Hertzian dipole, the normalized field pattern is given by

E_θn(θ, φ) = E_θ(r, θ, φ)

E_θmax(r) = sin θ = H_φn(θ, φ) (2.20)

z

y

x

1

Fig. 2.1 Normalized E_θ field pattern of a Hertzian dipole

A three-dimensional (3D) plot of the normalized electric field (maximum amplitude equal to unity) is shown in Fig. 2.1. The field intensity along a direction (θ, φ) is given by the length of the position vector to a point on the surface of the 3D shape in the direction (θ, φ).

The radiation pattern or antenna pattern is defined as ‘the spatial distribu-tion of a quantity that characterizes the electromagnetic field generated by an antenna’ (IEEE Std 145 1990). The quantity referred to in this definition could be the field amplitude, power, radiation intensity, antenna polariza-tion, relative phase, etc. When the term ‘radiation pattern’ is used with-out specifying the quantity, the radiation intensity or the field amplitude is implied.

It is generally convenient to present the pattern in two-dimensional (2D) plots by considering two orthogonal principal plane cuts of the 3D pattern. Principal plane implies that the cut is through the pat-tern maximum. For a linearly polarized antenna (see Section 2.4 for the definition), the pattern cuts in the principal planes parallel to the E and H field vectors are chosen. These patterns are called the E-plane and the H-plane patterns, respectively. For a Hertzian dipole, the x-z and x-y plane cuts of the pattern shown in Fig. 2.1 are the E-plane and the H-plane patterns and are shown in Fig. 2.2. The E-plane pattern resembles the shape of a figure-of-eight and the H-plane pattern is a circle.

It is also possible to define the 2D cuts in the spherical coordinate system. For example, the θ = 90^◦ cut is the same as the x-y cut. However, one half of the x-z cut is denoted by the φ = 0^◦ cut and the other half by the φ = 180^◦ cut.

180° (a)

 = 0°

30°

60°

90°

120°

150° 30°

60°

90°

120°

150°

1

0.4 0.8 E n0.6

0.2

150°

 = 0°

30°

60°

90°

120°

180°

(b) 330°

300°

270°

240°

210°

1

Eθ n 1.2

0.6 0.8

0.4 0.2

Fig. 2.2 Normalized E_θ field pattern of a Hertzian dipole: (a)E-plane pattern (x-zplane);

(b)H-plane pattern (x-y plane)

A 3D plot of the normalized power pattern expressed in decibels [Eqn (2.19)] is shown in Fig. 2.3(a) and the E-plane pattern is shown in Fig. 2.3(b). The pattern has a maximum along θ = 90^◦ which is also known as the broadside direction of the dipole. The plot indicates the relative level of the radiation intensity with respect to the maximum. For example, along θ = 30^◦ the radiation intensity is 6 dB below the maximum. The pattern has nulls along θ = 0^◦ and 180^◦, which are the directions along the axis of the dipole.

z

y

0

 = 0°

−20

−10

−30

30°

60°

90°

120°

180°

150° x

(b) (a)

Relative power (dB)

Fig. 2.3 Normalized power pattern of a Hertzian dipole (a) 3D view; (b)E-plane pattern (φ = 0^◦ plane) in dB units

The radiation pattern of an antenna can also have several other features.

Consider a three-dimensional normalized power pattern of an antenna as shown in Fig. 2.4. The direction of the radiation maximum is along (θ, φ) = (90^◦, 90^◦). The pattern indicates regions of higher radiation surrounded by regions of lower radiation. These are known as lobes. The lobe along the direction of maximum radiation is known as main lobe or major lobe and all other lobes are called side lobes. The main lobe is also sometimes referred to as the main beam. The nulls in the pattern indicate that along these directions the radiation is zero. The radiation lobe that makes an angle of about 180^◦ with the main lobe is known as the back lobe (Fig. 2.5).

The two-dimensional cut of a three-dimensional radiation pattern can also be plotted on a rectangular graph with the normalized power (in dB) along the vertical axis and the angle (θ) along the horizontal axis (Fig. 2.6). This representation is very useful for extracting quantitative information about the pattern. For example, we can infer that the first side lobe adjacent to the main lobe is 18 dB below the main lobe.

The angular width of the main beam between its half-power points is known as the half-power beamwidth or the 3 dB beamwidth. For patterns which are very broad, sometimes a 10 dB beamwidth is specified instead of a 3 dB beamwidth. The 10 dB beamwidth is the angle between the two directions on either side of the main lobe peak along which the power density

z

y

x

Fig. 2.4 Normalized power pattern of an antenna

is 10 dB below the maximum value. For the pattern shown in Fig. 2.6 the half-power beamwidth is 30^◦ and the 10 dB beamwidth is 50^◦.

The normalized pattern function, P_n(θ, φ), specifies the angular distri-bution of the total radiated power. For example, consider an antenna that radiates equally in all directions, i.e., the radiation pattern of the antenna is

180°

 = 0°

30°

60°

90°

120°

150°

30°

60°

90°

120°

150°

0

180°

 = 0°

30°

60°

90°

120°

150°

330°

300°

270°

240°

210°

0

−¹⁰

−²⁰

−³⁰

(a) (b)

Relative power (dB) Relative power (dB)

−²⁰

−¹⁰

−³⁰

Fig. 2.5 Polar plot of they-zandx-y cuts of the pattern shown in Fig. 2.4

0

−180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180

−3

−5

−40

−35

−30

−25

−20

−15

−10

Relative power (dB)

Angle (deg) First SLL Side lobes

10 dB BW 3 dB BW Main beam

Back lobe

Fig. 2.6 Rectangular plot of thex-y plane cut of the pattern shown in Fig. 2.4

a sphere. Such an antenna is known as an isotropic antenna and the corre-sponding radiation pattern is known as an isotropic pattern. The radiation pattern of an isotropic antenna is non-directional. The Hertzian dipole, on the other hand, has a non-directional pattern in one plane (x-y plane for a z-oriented dipole), but in any plane orthogonal to it, the pattern is direc-tional. Such a pattern is known as an omni-directional pattern. Several terms, such as, the pencil beam, fan beam, shaped beam, etc., are used in describing an antenna pattern based on the shape of the radiation pat-tern. A pencil beam antenna has maximum radiation in one direction and the beamwidths in the two orthogonal cuts of the pattern are small. If the beamwidth is broad in one cut and narrow in an orthogonal cut, the antenna pattern is called a fan-beam pattern.

EXAMPLE2.3

The electric field of an antenna is given by

E = a_θsin(4π cos θ) 4π cos θ

Calculate (a) the direction of the maximum, (b) the 3 dB beamwidth, (c) the direction and level of the first side lobe, and (d) the number of nulls in the pattern. Plot the power pattern on a rectangular graph.

Solution: The electric field has the form sin x/x, which has a maximum amplitude of 1 at x = 0, has an amplitude of 0.707 (which corresponds to the 3 dB point) at x = 1.39, second peak of−0.217 at x = 4.49, and nulls at x = nπ; n = 1, 2, . . ., where x = 4π cos θ. (See Appendix G for a description of the sin x/x function).

(a) The maximum of the pattern occurs along 4π cos θ_max= 0, which gives a value of θ_max= 90^◦.

(b) Let θ₁ be the direction along which the power is half of the maximum power. Since the sin x/x function has a value of 0.707 at x = 1.39, we have 1.39 = 4π cos θ₁ and, hence, θ₁ = cos⁻¹[1.39/(4π)] = 83.65^◦. Similarly, on the other side of the maximum, the direction θ₂ along which the electric field is equal to 0.707 of the maximum is 4π cos θ₂ =

−1.39, which gives, θ2 = 96.35^◦. The half-power beamwidth is Θ_HP= θ2− θ1= 12.7^◦.

(c) The direction of the first side lobe, θ_s, on either side of the main lobe is given by ±4.49 = 4π cos θs or θ_s= 69.07^◦ and 110.93^◦. The level of the first side lobe peak is 20 log₁₀(0.217) =−13.3 dB.

(d) The nth null of the pattern occurs at x = nπ. The direction of the nth null, θ_n is given by 4π cos θ_n = nπ or cos θ_n = n/4. This has real solutions only for n/4≤ 1. Therefore, the pattern has 4 nulls between θ = 0^◦ and θ = 90^◦ and 4 more nulls between θ = 90^◦ and θ = 180^◦ (see Fig. 2.7).

EXAMPLE2.4

Calculate the beamwidths in the x-y and y-z planes of an antenna the power pattern of which is given by

U (θ, φ) =

sin²θ sin φ 0≤ θ ≤ π; 0 ≤ φ ≤ π 0 0≤ θ ≤ π; π ≤ φ ≤ 2π

Solution: In the x-y plane, θ = π/2 and the power pattern is given by U

π 2, φ



= sin φ

0

0 30 60 90 120 150 180

−3

−5

−40

−35

−30

−25

−20

−15

−10

Relative power (dB)

Angle (deg) 13.3 dB

3 dB beamwidth

Main lobe

First side lobe

Fig. 2.7 Radiation pattern of the antenna of Example 2.3;P_ndB(θ, φ) = 20 log₁₀[sin(4π cos θ)/(4π cos θ)]

The angles along which the power is half the maximum value (3 dB below the maximum) is given by the solutions of

sin φ = 0.5

which is satisfied for φ = 30^◦ and φ = 150^◦. Therefore, the 3 dB beamwidth in the x-y plane is 150^◦− 30^◦ = 120^◦.

In the y-z plane (φ = π/2), the power pattern is given by U

 θ,π

2



= sin²θ

and the 3 dB points occur along θ satisfying the condition sin²θ = 0.5

which gives the values of θ as 45^◦ and 135^◦. Therefore, the beamwidth in the y-z plane is 90^◦.

EXAMPLE2.5

Derive an expression for the time-averaged power density vector of the elec-tromagnetic wave radiated by a Hertzian dipole of length dl, kept at the origin, oriented along the x-axis, and excited by a current of amplitude I₀. Solution: The far-fields of a Hertzian dipole oriented along the x-direction are (see Example 1.8)

H_θ =−jkI0dl

4π sin φe^−jkr r H_φ=−jkI0dl

4π cos θ cos φe^−jkr r E_θ =−jηkI0dl

4π cos θ cos φe^−jkr r E_φ= jηkI0dl

4π sin φe^−jkr r The time-averaged power density is given by

S = 1

2Re{E × H^∗}

Expressing the field vectors in terms of their components S = 1

2Re{(aθE_θ+ a_φE_φ)× (aθH_θ+ a_φH_φ)^∗}

Expanding the cross product using the identities a_θ× aθ = 0, a_φ× aφ= 0, a_θ× aφ= a_r, and a_φ× aθ =−ar

S = 1

2Re{arE_θH_φ^∗− arE_φH_θ^∗} Substituting the field expressions and simplifying

S = a_r1 2η

k|I0| dl 4πr

2

cos²θ cos²φ + sin²φ

A 3D plot of the radiation intensity as a function of θ and φ is shown in Fig. 2.8. The radiation pattern is identical to that of a z-directed dipole but rotated by 90^◦.

z

y x

Fig. 2.8 Normalized power pattern of an x-directed Hertzian dipole

In document [Harish, A.R. Sachidananda, M.] Antennas and Wave (Page 43-55)