GAIN

Fundamental Parameters of Antennas

2.9 GAIN

Another useful measure describing the performance of an antenna is thegain. Although the gain of the antenna is closely related to the directivity, it is a measure that takes into

account the efficiency of the antenna as well as its directional capabilities. Remember that directivity is a measure that describes only the directional properties of the antenna, and it is therefore controlled only by the pattern.

Gainof an antenna (in a given direction) is defined as “the ratio of the intensity, in a given direction, to the radiation intensity that would be obtained if the power accepted by the antenna were radiated isotropically. The radiation intensity corresponding to the isotropically radiated power is equal to the power accepted (input) by the antenna divided by 4π.” Inequationform this canbe expressed as

Gain=4π radiation intensity

total input (accepted) power =4πU (θ, φ)

P_in (dimensionless) (2-46) Inmost cases we deal with relative gain, which is defined as “the ratio of the power gain in a given direction to the power gain of a reference antenna in its refer- enced direction.” The power input must be the same for both antennas. The reference antenna is usually a dipole, horn, or any other antenna whose gain can be calculated or it is known. In most cases, however, the reference antenna is a lossless isotropic source. Thus

G= 4π U (θ, φ)

P_in(lossless isotropic source) (dimensionless) (2-46a) When the direction is not stated, the power gain is usually taken in the direction of maximum radiation.

Referring to Figure 2.22(a), we can write that the total radiated power (Prad) is related to the total input power(Pin)by

P_rad=e_cdP_in (2-47)

wheree_cd is the antenna radiation efficiency (dimensionless) which is defined in (2-44), (2-45) and Section 2.14 by (2-90). According to the IEEE Standards, “gain does not include losses arising from impedance mismatches (reflection losses) and polarization mismatches (losses).”

In this edition of the book we define two gains; one, referred to as gain (G), and the other, referred to as absolute gain (Gabs), that also takes into account the reflection/mismatch losses represented in both (2-44) and (2-45).

Using (2-47) reduces (2-46a) to G(θ, φ)=ecd

4πU (θ, φ) P_rad

(2-48) which is related to the directivity of (2-16) and (2-21) by

G(θ, φ)=ecdD(θ, φ) (2-49)

In a similar manner, the maximum value of the gain is related to the maximum direc- tivity of (2-16a) and (2-23) by

G0=G(θ, φ)|max=ecdD(θ, φ)|max=ecdD0 (2-49a)

GAIN 67

While (2-47) does take into account the losses of the antenna element itself,it does not take into account the losses when the antenna element is connected to a transmis- sion line, as shown in Figure 2.22. These connection losses are usually referred to as reflections (mismatch) losses, and they are taken into account by introducing a reflec- tion(mismatch) efficiencyer, which is related to the reflectioncoefficient as shownin (2-45) orer =(1− |?|²). Thus, we canintroduce anabsolute gainG_abs that takes into account the reflection/mismatch losses (due to the connection of the antenna element to the transmission line), and it can be written as

G_abs(θ, φ)=e_rG(θ, φ)=(1− |?|²)G(θ, φ)

=erecdD(θ, φ)=eoD(θ, φ) (2-49b) whereeo is the overall efficiency as defined in (2-44), (2-45). Similarly, themaximum absolute gainG_0abs of (2-49a) is related to the maximum directivityD₀ by

G0abs=Gabs(θ, φ)|max=erG(θ, φ)|max=(1− |?|²)G(θ, φ)|max

=e_re_cdD(θ, φ)|max=e_oD(θ, φ)|max=e_oD₀

(2-49c)

If the antenna is matched to the transmission line, that is, the antenna input impedance Zinis equal to the characteristic impedanceZcof the line(|?| =0), then the two gains are equal(Gabs =G).

As was done with the directivity, we can define the partial gain of an antenna for a given polarization in a given direction as “that part of the radiation intensity corresponding to a given polarization divided by the total radiation intensity that would be obtained if the power accepted by the antenna were radiated isotropically.” With this definition for the partial gain, then, in a given direction, “the total gain is the sum of the partial gains for any two orthogonal polarizations.” For a spherical coordinate system, the total maximum gainG₀ for the orthogonalθ andφ components of an antenna can be written, in a similar form as was the maximum directivity in (2-17)–(2-17b), as

G₀ =Gθ+Gφ (2-50)

while the partial gainsG_θ andG_φ are expressed as Gθ = 4π U_θ

Pin

(2-50a) G_φ = 4π U_φ

P_in (2-50b)

where

Uθ =radiationintensity ina givendirectioncontained inEθ field component U_φ =radiationintensity ina givendirectioncontained inE_φ field component P_in =total input (accepted) power

For many practical antennas an approximate formula for the gain, corresponding to (2-27) or (2-27a) for the directivity, is

G₀ 30,000

4_1d4_2d (2-51)

In practice, whenever the term “gain” is used, it usually refers to themaximum gain as defined by (2-49a) or (2-49c).

Usually the gain is given in terms of decibels instead of the dimensionless quantity of (2-49a). The conversionformula is givenby

G₀(dB)=10 log₁₀[e_cdD₀ (dimensionless)] (2-52)

Example 2.10

A lossless resonant half-wavelength dipole antenna, with input impedance of 73 ohms, is connected to a transmission line whose characteristic impedance is 50 ohms. Assuming that the pattern of the antenna is given approximately by

U=B₀sin³θ find the maximum absolute gain of this antenna.

Solution: Let us first compute the maximum directivity of the antenna. For this

U|max=Umax=B0

Prad= 2π

0

π 0

U (θ, φ)sinθ dθ dφ=2π B0

π 0

sin⁴θ dθ=B0

3π² 4

D0=4πUmax

Prad = 16

3π =1.697

Since the antenna was stated to be lossless, then the radiation efficiency ecd=1.

Thus, the total maximum gainis equal to

G0=ecdD0=1(1.697)=1.697 G0(dB)=10 log₁₀(1.697)=2.297 which is identical to the directivity because the antenna is lossless.

There is another loss factor which is not taken into account in the gain. That is the loss due to reflection or mismatch losses between the antenna (load) and the transmission line.

This loss is accounted for by the reflection efficiency of (2-44) or (2-45), and it is equal to

er=(1− |?|²)=

1− 73−50

73+50 ²

=0.965 er(dB)=10 log₁₀(0.965)= −0.155

BEAM EFFICIENCY 69

Therefore the overall efficiency is

e0=erecd=0.965 e₀(dB)= −0.155

Thus, the overall losses are equal to 0.155 dB. The absolute gainis equal to G_0abs =e₀D₀=0.965(1.697)=1.6376 G0abs(dB)=10 log₁₀(1.6376)=2.142

The gain in dB can also be obtained by converting the directivity and radiation efficiency indB and thenadding them. Thus,

ecd(dB)=10 log₁₀(1.0)=0 D0(dB)=10 log₁₀(1.697)=2.297 G0(dB)=ecd(dB)+D0(dB)=2.297

which is the same as obtained previously. The same procedure can be used for the absolute gain.