# 9.2 Antenna Radiation Characteristics

An antenna pattern describes the far-field directional properties of an antenna when measured at a fixed distance from the antenna. In general, the antenna pattern is a three-dimensional plot that displays the strength of the radiated field or power density as a function of direction—with direction being specified by the zenith angle θ and the azimuth angle ϕ.

Consider a transmitting antenna placed at the origin of the observation sphere shown in Fig. 9-8. The differential power radiated by the antenna through an elemental area d A is

d P rad=S av⋅d A=S av⋅R ˆ d A=S d A(W),(9.16)

where S is the radial component of the time-average Poynting vector S av. In the far-field region of any antenna, S av is always in the radial direction. In a spherical coordinate system,

d A=R 2 sin θ d θ d ϕ,(9.17)

and the solid angle d Ω associated with d A, which is defined as the subtended area divided by R 2, is given by

Note that, whereas a planar angle is measured in radians and the angular measure of a complete circle is 2 π (rad), a solid angle is measured in steradians (sr), and the angular measure for a spherical surface is Ω=(4 π R 2)/R 2=4 π (sr). The solid angle of a hemisphere is 2 π (sr).

# Figure 9-8 Definition of solid angle d Ω= sin θ d θ d ϕ.

Figure 9 8 Full Alternative Text

Using the relation d A=R 2 d Ω, d P rad can be rewritten as

d P rad=R 2 S(R,θ,ϕ)d Ω.(9.19)

The total power radiated by an antenna through a spherical surface at a fixed distance R is obtained by integrating Eq. (9.19) over that surface:

P rad=R 2∫ϕ=0 2 π∫θ=0 π S(R,θ,ϕ) sin θ d θ d ϕ=R 2 S max∫ϕ=0 2 π∫θ=0 π F(θ,ϕ) sin θ d θ d ϕ=R 2 S max∬4 π F(θ,ϕ)d Ω(W),(9.20)

where F(θ,ϕ) is the normalized radiation intensity defined by Eq. (9.13). The 4 π symbol under the integral sign is used as an abbreviation for the indicated limits on θ and ϕ. Formally, P rad is called the total radiated power.

# 9.2.1 Antenna Pattern

Each specific combination of the zenith angle θ and the azimuth angle ϕ denotes a specific direction in the spherical coordinate system of Fig. 9-8. The normalized radiation intensity F(θ,ϕ) characterizes the directional pattern of the energy radiated by an antenna, and a plot of F(θ,ϕ) as a function of both θ and ϕ constitutes a three-dimensional pattern—an example of which is shown in Fig. 9-9.

# Figure 9-9 Three-dimensional pattern of a narrow-beam antenna.

Figure 9 9 Full Alternative Text

Often, it is of interest to characterize the variation of F(θ,ϕ) in the form of two-dimensional plots in specific planes in the spherical coordinate system. The two planes most commonly specified for this purpose are the elevation and azimuth planes. The elevation plane, also called the θ plane, is a plane corresponding to a constant value of ϕ. For example, ϕ=0 defines the x–z plane and ϕ=90° defines the y–z plane, both of which are elevation planes (Fig. 9-8). A plot of F(θ,ϕ) versus θ in either of these planes constitutes a two-dimensional pattern in the elevation plane. This is not to imply, however, that the elevation-plane pattern is necessarily the same in all elevation planes.

The azimuth plane, also called the ϕ plane, is specified by θ=90° and corresponds to the x–y plane. The elevation and azimuth planes are often called the two principal planes of the spherical coordinate system.

Some antennas exhibit highly directive patterns with narrow beams, in which case it is often convenient to plot the antenna pattern on a decibel scale by expressing F in decibels:

F (dB)=10 log F.

As an example, the antenna pattern shown in Fig. 9-10(a) is plotted on a decibel scale in polar coordinates with its intensity as the radial variable. This format permits a convenient visual interpretation of the directional distribution of the radiation lobes.

# Figure 9-10 Representative plots of the normalized radiation pattern of a microwave antenna in (a) polar form and (b) rectangular form.

(a) Polar diagram

Figure 9 10-a Full Alternative Text

(b) Rectangular plot

Figure 9 10-b Full Alternative Text

Another format commonly used for inspecting the pattern of a narrow-beam antenna is the rectangular display shown in Fig. 9-10(b), which permits the pattern to be easily expanded by changing the scale of the horizontal axis. These plots represent the variation in only one plane in the observation sphere: the ϕ=0 plane. Unless the pattern is symmetrical in ϕ, additional patterns are required to define the overall variation of F(θ,ϕ) with θ and ϕ.

Strictly speaking, the polar angle θ is always positive, being defined over the range from 0° (z direction) to 180° (−z direction), yet the θ axis in Fig. 9-10(b) is shown to have both positive and negative values. This is not a contradiction, but rather a different form of plotting antenna patterns. The right-hand half of the plot represents the variation of F (dB) with θ because θ is increased in a clockwise direction in the x–z plane [see inset in Fig. 9-10(b)], corresponding to ϕ=0. The left-hand half of the plot represents the variation of F (dB) with θ as θ is increased in a counterclockwise direction at ϕ=180°. Thus, a negative θ value simply denotes that the direction (θ,ϕ) is in the left-hand half of the x–z plane.

The pattern shown in Fig. 9-10(a) indicates that the antenna is fairly directive, since most of the energy is radiated through a narrow sector called the main lobe. In addition to the main lobe, the pattern exhibits several side lobes and back lobes as well. For most applications, these extra lobes are considered undesirable because they represent wasted energy for transmitting antennas and potential interference directions for receiving antennas.

# 9.2.2 Beam Dimensions

For an antenna with a single main lobe, the pattern solid angle Ω p describes the equivalent width of the main lobe of the antenna pattern (Fig. 9-11). It is defined as the integral of the normalized radiation intensity F(θ,ϕ) over a sphere:

# Figure 9-11 The pattern solid angle Ω p defines an equivalent cone over which all of the radiation of the actual antenna is concentrated with uniform intensity equal to the maximum of the actual pattern.

(a) Actual pattern

(b) Equivalent solid angle

The pattern solid angle characterizes the directional properties of the three-dimensional radiation pattern. To characterize the width of the main lobe in a given plane, the term used is beamwidth. The half-power beamwidth, or simply the beamwidth β, is defined as the angular width of the main lobe between the two angles at which the magnitude of F(θ,ϕ) is equal to half of its peak value (or −3 dB on a decibel scale). For example, for the pattern displayed in Fig. 9-10(b), β is given by

β=θ 2−θ 1,(9.22)

where θ 1 and θ 2 are the half-power angles at which F(θ,0)=0.5 (with θ 2 denoting the larger value and θ 1 denoting the smaller one, as shown in the figure). If the pattern is symmetrical and the peak value of F(θ,ϕ) is at θ=0, then β=2 θ 2. For the short-dipole pattern shown earlier in Fig. 9-7(a), F(θ) is maximum at θ=90°, θ 2 is at 135°, and θ 1 is at 45°. Hence, β=135°−45°=90°. The beamwidth β is also known as the 3 dB beamwidth. In addition to the half-power beamwidth, other beam dimensions may be of interest for certain applications, such as the null beamwidth β null, which is the angular width between the first nulls on the two sides of the peak (Fig. 9-10(b)).

# 9.2.3 Antenna Directivity

The directivity D of an antenna is defined as the ratio of its maximum normalized radiation intensity, F max (which by definition is equal to 1), to the average value of F(θ,ϕ) over all directions (4 π space):

D=F max F av=1 1 4 π∬4 π F(θ,ϕ)d Ω=4 π Ω p.(dimensionless)(9.23)

Here Ω p is the pattern solid angle defined by Eq. (9.21). Thus, the narrower Ω p of an antenna pattern is, the greater is the directivity. For an isotropic antenna, Ω p=4 π; hence, its directivity D iso=1.

By using Eq. (9.20) in Eq. (9.23), D can be expressed as

D=4 π R 2 S max P rad=S max S av,(9.24)

where S av=P rad/(4 π R 2) is the average value of the radiated power density and is equal to the total power radiated by the antenna, P rad, divided by the surface area of a sphere of radius R.

Usually, D is expressed in decibels:* D (dB)=10 log D.

For an antenna with a single main lobe pointing in the z direction shown in Fig. 9-12, Ω p may be approximated as the product of the half-power beamwidths β x z and β y z (in radians):

Ω p≈β x z β y z,(9.25)

and

# Figure 9-12 The solid angle of a unidirectional radiation pattern is approximately equal to the product of the half-power beamwidths in the two principal planes; that is, Ω p≈β x z β y z.

Figure 9 12 Full Alternative Text

Although approximate, this relation provides a useful method for estimating the antenna directivity from measurements of the beamwidths in the two orthogonal planes whose intersection is the axis of the main lobe.

# Example 9-1: Antenna Radiation Properties

Determine (a) the direction of maximum radiation, (b) pattern solid angle, (c) directivity, and (d) half-power beamwidth in the y–z plane for an antenna that radiates only into the upper hemisphere with normalized radiation intensity given by F(θ,ϕ)=cos 2 θ.

Solution: The statement that the antenna radiates through only the upper hemisphere is equivalent to

F(θ,ϕ)=F(θ)={cos 2 θ for 0≤θ≤π/2 and 0≤ϕ≤2 π,0 elsewhere.

(a) The function F(θ)=cos 2 θ is independent of ϕ and is maximum when θ=0°. A polar plot of F(θ) is shown in Fig. 9-13.

# Figure 9-13 Polar plot of F(θ)=cos 2 θ.

Figure 9 13 Full Alternative Text

(b) From Eq. (9.21), the pattern solid angle Ω p is given by

Ω p=∬4 π F(θ,ϕ)d Ω=∫ϕ=0 2 π∫θ=0 π/2 cos 2 θ sin θ d θ d ϕ=∫ϕ=0 2 π−cos 3 θ 3 0 π/2 d ϕ=∫0 2 π 1 3 d ϕ=2 π 3(sr).

(c) Application of Eq. (9.23) gives

D=4 π Ω p=4 π 3 2 π=6,

which corresponds to D (dB)=10 log 6=7.78 dB.

(d) The half-power beamwidth β is obtained by setting F(θ)=0.5. That is,

F(θ)=cos 2 θ=0.5,

which gives the half-power angles θ 1=−45° and θ 2=45°. Hence,

β=θ 2−θ 1=90°.

# Example 9-2: Directivity of a Hertzian Dipole

Calculate the directivity of a Hertzian dipole.

Solution: Application of Eq. (9.23) with F(θ)=sin 2 θ [from Eq. (9.15)] gives

D=4 π∬4 π F(θ,ϕ) sin θ d θ d ϕ=4 π∫ϕ=0 2 π∫θ=0 π sin 3 θ d θ d ϕ=4 π 8 π/3=1.5

or, equivalently, 1.76 dB.

# 9.2.4 Antenna Gain

Of the total power P t (transmitter power) supplied to the antenna, a part, P rad, is radiated out into space, and the remainder, P loss, is dissipated as heat in the antenna structure. The radiation efficiency ξ is defined as the ratio of P rad to P t:

The gain of an antenna is defined as

G=4 π R 2 S max P t,(9.28)

which is similar in form to the expression given by Eq. (9.24) for the directivity D except that it is referenced to the input power supplied to the antenna, P t, rather than to the radiated power P rad. In view of Eq. (9.27),

# 9.2.5 Radiation Resistance

To a transmission line connected between a generator supplying power P t on one end and an antenna on the other end, the antenna is merely a load with input impedance Z in. If the line is lossless and properly matched to the antenna, all of P t is transferred to the antenna. In general, Z in consists of a resistive component R in and a reactive component X in:

Z in=R in+j X in.(9.30)

The resistive component is defined as equivalent to a resistor R in that would consume an average power P t when the amplitude of the ac current flowing through it is I 0,

P t=1 2 I 0 2 R in.(9.31)

Since P t=P rad+P loss, it follows that R in can be defined as the sum of a radiation resistance R rad and a loss resistance R loss,

R in=R rad+R loss,(9.32)

with

P rad=1 2 I 0 2 R rad, (9.33a)
P loss=1 2 I 0 2 R loss, (9.33b)

where I 0 is the amplitude of the sinusoidal current exciting the antenna. As defined earlier, the radiation efficiency is the ratio of P rad to P t, or

The radiation resistance R rad can be calculated by integrating the far-field power density over a sphere to obtain P rad and then equating the result to Eq. (9.33a).

# Example 9-3: Radiation Resistance and Efficiency of a Hertzian Dipole

A 4-cm long center-fed dipole is used as an antenna at 75 MHz. The antenna wire is made of copper and has a radius a=0.4 m m. From Eqs. (7.92a) and (7.94), the loss resistance of a circular wire of length l is given by

R loss=1 2 π a π f μ c σ c,(9.35)

where μ c and σ c are the magnetic permeability and conductivity of the wire, respectively. Calculate the radiation resistance and the radiation efficiency of the dipole antenna.

Solution: At 75 MHz,

λ=c f=3×10 8 7.5×10 7=4 m.

The length to wavelength ratio is l/λ=4 cm/4 m=10−2. Hence, this is a short dipole. From Eq. (9.24),

P rad=4 π R 2 D S max.(9.36)

For the Hertzian dipole, S max is given by Eq. (9.14), and from Example 9-2, we established that D=1.5. Hence,

P rad=4 π R 2 1.5×15 π I 0 2 R 2 l λ 2=40 π 2 I 0 2 l λ 2.(9.37)

Equating this result to Eq. (9.33a) and then solving for the radiation resistance R rad leads to

For l/λ=10−2, R rad=0.08 Ω.

Next, we determine the loss resistance R loss. For copper, Appendix B gives μ c≈μ 0=4 π×10−7 H/m and σ c=5.8×10 7 S/m. Hence,

R loss=l 2 π a π f μ c σ c,=4×10−2 2 π×4×10−4(π×75×10 6×4 π×10−7 5.8×10 7)1/2=0.036 Ω.

Therefore, the radiation efficiency is

ξ=R rad R rad+R loss=0.08 0.08+0.036=0.69.

Thus, the dipole is 69% efficient.

Concept Question 9-5: What does the pattern solid angle represent?

Concept Question 9-6: What is the magnitude of the directivity of an isotropic antenna?

Concept Question 9-7: What physical and material properties affect the radiation efficiency of a fixed-length Hertzian dipole antenna?