# ANTENNA PERFORMANCE MEASURES

Antennas are devices that are used in systems for communi­cations or sensing. There are many parameters used to quan­tify the performance of the antenna as a device, which in turn impacts on system performance. In this section we consider the most important of these parameters when they are em­ployed in their primary application area of communication links, such as the simple communication link as shown in Fig. 9. We first discuss the basic properties of a receiving antenna.

 Transmitter

 R

 where |F(max)| is the maximum value of the pattern magni­tude and |F(SLL)| is the pattern value of the maximum of the highest side lobe magnitude. For a normalized pattern F(max) = 1. The width of the main beam is quantified through half­power beamwidth, HP, which is the angular separation of the points where the main beam of the power pattern equals one-
 matched to the wave, and is impedance matched to its load. The maximum refers to the assumption that there are no ohmic losses on the antenna. Maximum effective aperture for the ideal dipole is found using Eqs. (92) and (93) with Eq. (94) to give

 (a)

 (Ь)

 Figure 10. Equivalent circuit for a receiving antenna. (a) Receive antenna connected to a receiver with load impedance ZL. (b) Equiva­lent circuit.

 W 8 Rr TiW 2 г}

 P л _ Am S

 1 n 2 3 2 = 4й:(Аг) =8^л

 (95)

 where the ideal dipole radiation resistance value of [2w/3 ^(Дz/A)2] was used. The maximum effective aperture of an ideal dipole is independent of its length Az (as long as Az < A). However, it is important to note that Rr is proportional to (Az/A)2 so that even though Aem remains constant as the dipole is shortened, its radiation resistance decreases rapidly and it is more difficult to realize this maximum effective aperture because of the required conjugate impedance match of the re­ceiver to the antenna. The directivity of the ideal dipole can be written in the following manner:

 The receiving antenna with impedance ZA and terminated in load impedance ZL is modeled as shown in Fig. 10. The total power incident on the receiving antenna is found by summing up the incident power density over the area of the receive antenna, called effective aperture. How an antenna converts this incident power into available power at its terminals de­pends on the type of antenna used, its pointing direction, and polarization. In this section we discuss the basic relationships for power calculations and illustrate their use in communica­tion links. Directivity and Gain. For system calculations it is usually easier to work with directivity rather than its equivalent, maximum effective aperture. The relation can be established by examining an infinitestimal dipole and generalizing. The maximum effective aperture of an ideal, lossless dipole of length Дг is found by orienting the dipole for maximum re­sponse, which is parallel to the incoming linearly polarized electric field E’. Then the open circuit voltage is found from

 3 4n 3 2 ~2~ І2"8яЛ

 Ideal dipole

 (96)

 Grouping factors this way permits identification of Ael Eq. (95). Thus D — —A u — ^2 ™

 from

 (97)

 Although we derived this for an ideal dipole, this relationship is true for any antenna. For an isotropic antenna, the directiv­ity by definition is unity; so from Eq. (97) with D = 1

 Aem = — Isotropic antenna 4n

 (98)

 VA = E’ Az Ideal dipole receiving antenna

 (91)
 (99)
 X

The power available from the antenna is realized when the antenna impedance is matched by a load impedance of ZL = Rr — jXA assuming Rohmic = 0. Rr is the radiation resistance. The maximum available power is then

Comparing this to the definition of directivity in (100) below we see that

2 — A О — л em0A

 1 |E’|2

 (Az)2

 (92)

 Rr

 (93)

 The available power is found using the maximum effective ap­erture Aem, which is the collecting area of the antenna. The receiving antenna collects power from the incident wave in proportion to its maximum effective aperture

 where Eq. (91) was used. The available power can also be calculated by examining the incident wave. The power density (Poynting vector magnitude) in the incoming wave is

 4 1{El{2 2 2 ~rj

 ^Am — ~

 1 |Va I2

 Rr

 which is also a general relationship. We can extract some in­teresting concepts from this relation. For a fixed wavelength Aem and HA, are inversely proportional; that is, as the maxi­mum effective aperture increases (as a result of increasing its physical size), the beam solid angle decreases, which means power is more concentrated in angular space (i. e., directivity goes up). For a fixed maximum effective aperture (i. e., an­tenna size), as wavelength decreases (frequency increases) the beam solid angle also decreases, leading to increased di­rectivity. Directivity is more directly related to its definition through this inverse dependence on beam solid angle as

 (100)

 fix
 (94)
 where

PAm = ®Aem

 Oa = И IF(в, ф)І2 dO
 (101)

The maximum available power PAm will be realized if the an­tenna is directed for maximum response, is polarization

 This directivity definition has a simple interpretation. Direc­tivity is a measure of how much greater the power density at a fixed distance is in a given direction than if all power were radiated isotropically. This view is illustrated in Fig. 11. For an isotropic antenna, as in Fig. 11(a), the beam solid angle is 4w, and thus Eq. (100) gives a directivity of unity. In practice antennas are not completely lossless. Earlier we saw that power available at the terminals of a transmit­ting antenna was not all transformed into radiated power. The power received by a receiving antenna is reduced to the fraction er (radiation efficiency) from what it would be if the antenna were lossless. This is represented by defining effec­tive aperture

 If the transmitting antenna were isotropic, it would have power density at distance R of

 Uave ~w

 S =

 (106)

 4n R2

 where Pt is the time-averaging input power (Pin) accepted by the transmitting antenna. The quantity Uave denotes the time average radiation intensity given in the units of power per solid angle see Fig. 11. For a transmitting antenna that is not isotropic but has gain Gt and is pointed for maximum power density in the direction of the receiver, we have for the power density incident on the receiving antenna,

 8=GtUt

 GA 4 nR2

 (107)

 Ae — erAe

 (102)

 R2
 (103)
 GtPtAer

and the available power with antenna losses included, analo­gous to Eq. (94), is

Pa — SAe

Using this in Eq. (103) gives the available received power as

(108)

Pr — SAer — 4яД2

 (109)

This simple equation is very intuitive and indicates that a receiving antenna acts to convert incident power (flux) den­sity in W/m2 to power delivered to the load in watts. Losses associated with mismatch between the polarization of the in­cident wave and receiving antenna as well as impedance mis­match between the antenna and load are not included in Ae. These losses are not inherent to the antenna, but depend on how it is used in the system. The concept of gain is introduced to account for losses on an antenna, that is, G = eD. We can form a gain expression from the directivity expression by mul­tiplying both sides of Eq. (97) by er and using Eq. (102): where Aer is the effective aperture of the receiving antenna and we assume it to be pointed and polarized for maximum response. Now from Eq. (104) Aer = GrA2/4w, so Eq. (108) be­comes

„ „ GtGTX2

P — P

r ‘ (4лR)2

which gives the available power in terms of the transmitted power, antenna gains, and wavelength. Or, we could use Gt = 4wAet/A2 in Eq. (108) giving

 p _ p AetAer r ‘ R2k2

 „ „ 4п 4я Сг — erL) — ~^2 ^r em —

 (110)

 (104)
 (105)

 Figure 11. Illustration of directivity. (a) Radiation intensity distrib­uted isotropically. (b) Radiation intensity from an actual antenna.

 where

 Ua

 (a)

 (111)

 PD = power delivered from the antenna Pr = power available from the receiving antenna p = polarization efficiency (or polarization mismatch fac­tor), 0 < p < 1 q = impedance mismatch factor, 0 < q < 1

 It is important to note that although we developed the gen­eral relationships of Eqs. (97), (99), and (104) for receiving antennas, they apply to transmitting antennas as well. The relationships are essential for communication system compu­tations that we consider next. Communication Links. We are now ready to completely de­scribe the power transfer in the communication link of Fig. 9.

 For electrically large antennas effective aperture is equal to or less than the physical aperture area of the antenna Ap, which is expressed using aperture efficiency eap:

 Ae — ^apAp

 which is called the Friis transmission formula (2). The power transmission formula Eq. (109) is very useful for calculating signal power levels in communication links. It assumes that the transmitting and receiving antennas are matched in impedance to their connecting transmission lines, have identical polarizations, and are aligned for polarization match. It also assumes the antennas are pointed toward each other for maximum gain. If any of these conditions are not met, it is a simple matter to correct for the loss introduced by polarization mismatch, impedance mismatch, or antenna misalignment. The antenna misalignment effect is easily included by us­ing the power gain value in the appropriate direction. The effect and evaluation of polarization and impedance mismatch are additional considerations. Figure 10 shows the network model for a receiving antenna with input antenna impedance ZA and an attached load impedance ZL, which can be a trans­mission line connected to a distant receiver. The power deliv­ered to the terminating impedance is

 Pd — pqPr
 Figure 12. Antenna temperature. (a) An antenna receiving noise from directions (в, ф) producing antenna temperature TA. (b) Equiva­lent model.
 (a) (b)

An overall efficiency, or total efficiency etotal, can be defined to include the effects of polarization and impedance mismatch:

^ total = pqe ap (112)

Then PD = etotaiPr. It is convenient to express Eq. (111) in dB form:

PD (dBm) = 10log p + 10log q + Pr(dBm) (113)

where the unit dBm is power in decibels above a milliwatt; for example, 30 dBm is 1 W. Both powers could also be ex­pressed in units of decibels above a watt, dBW. The power transmission formula Eq. (109) can also be expressed in dB form as

Pr (dBm) = Pt(dBm) + Gt (dB) + Gr (dB)

— 20log R(km) — 20 log f(MHz) — 32.44 (114)

where Gt(dB) and Gr(dB) are the transmit and receive an­tenna gains in decibels, R(km) is the distance between the transmitter and receiver in kilometers, and f(MHz) is the fre­quency in megahertz.

Effective Isotropically Radiated Power. A frequently used concept in communication systems is that of effective (or equivalent) isotropically radiated power, EIRP. It is formally defined as the power gain of a transmitting antenna in a given direction multiplied by the net power accepted by the antenna from the connected transmitter. Sometimes it is de­noted as ERP, but this term, effective radiated power, is usu­ally reserved for EIRP with antenna gain relative to that of a half-wave dipole instead of gain relative to an isotropic an­tenna. As an example of EIRP, suppose an observer is located in the direction of maximum radiation from a transmitting antenna with input power Pt. Then the EIRP may be ex­pressed as

EIRP = PtGt (115)

For a radiation intensity Um, as illustrated in Fig. 11(b), and Gt = 4wUm/Pt, we obtain

EIRP = Pt= 4 7tUm (116)

Pt

The same radiation intensity could be obtained from a loss­less isotropic antenna (with power gain Gi = 1) if it had an input power Рід equal to PtGt. In other words, to obtain the same radiation intensity produced by the directional antenna in its pattern maximum direction, an isotropic antenna would have to have an input power Gt times greater. Effective iso — tropically radiated power is a frequently used parameter. For example, FM radio stations often mention their effective radi­ated power when they sign off at night.

Receiving systems are vulnerable to noise and a major contri­bution is the receiving antenna, which collects noise from its surrounding environment. In most situations a receiving an­tenna is surrounded by a complex environment as shown in

Fig. 12(a). Any object (except a perfect reflector) that is above absolute zero temperature will radiate electromagnetic waves. An antenna picks up this radiation through its an­tenna pattern and produces noise power at its output. The equivalent terminal behavior is modeled in Fig. 12(b) by con­sidering the radiation resistance of the antenna to be a noisy resistor at a temperature TA such that the same output noise power from the antenna in the actual environment is pro­duced. The antenna temperature TA is not the actual physical temperature of the antenna, but is an equivalent temperature that produces the same noise power, PNA, as the antenna op­erating in its surroundings. This equivalence is established by assuming the model of Fig. 12(b); the noise power available from the noise resistor in bandwidth Af at temperature TA is

Pna = kTKA f (117)

where

PNA = available power due to antenna noise [W] k = Boltzmann’s constant = 1.38 X 10—23 JK—1 TA = antenna temperature [K]

Such noise is often referred to as Nyquist or Johnson noise for system calculations. The system noise power PN is calcu­lated using the total system noise temperature Tsys in place of TA in Eq. (117) with Tsys = TA + Tr, where Tr is the receiver noise temperature.

Antenna noise is important in several system applications including communications and radiometry. Communication systems are evaluated through carrier-to-noise ratio, which is determined from the signal power and the system noise power as

P

CNR=^ (118)

PN

where PN = kTsys Af is the system noise power. This noise power equals the sum of PNA and noise power generated in the receiver connected to the antenna.

Noise power is found by first evaluating antenna tempera­ture. As seen in Fig. 12(a), TA is found from the collection of

 noise through the scene temperature distribution T(Q, ф) weighted by the response function of the antenna, the normal­ized power pattern P(0, ф). This is expressed mathematically by integrating over the temperature distribution:

 of technology development, including new antenna designs. DirecTv (trademark of Hughes Network Systems) transmits from 12.2 to 12.7 GHz with 120 W of power and an EIRP of about 55 dBW in each 24 MHz transponder that handles sev­eral compressed digital video channels. The receiving system uses a 0.46 m (18 in) diameter offset fed reflector antenna. In this example we perform the system calculations using the following link parameter values: f — 12.45 GHz (midband) Pt(dBW) — 20.8dBW (120 W) Gt(dB) — EIRP(dBW) — Pt(dBW) — 55 — 20.8 — 34.2 dB R — 38, 000 km (typical slant path length)

 і ґ ґ •’0 J0

 TA = ^r

 T(в, ф)Р(в, ф) dQ

 (119)

 If the scene is of constant temperature To over all angles, To comes out of the integral and then

 2n t P(e^)dn= ~^nA = T0 Q Д

 T _ T° 1a~o~

 ’41 ‘A J0 J0

 (120)

using Eq. (101) for ПА. The antenna is completely surrounded by noise of temperature To and its output antenna tempera­ture equals To independent of the antenna pattern shape.

In general, antenna noise power PNA is found from Eq. (117) using TA from Eq. (119) once the temperature distribu­tion T(0, ф) is determined. Of course, this depends on the scene, but in general T(0, ф) consists of two components: sky noise and ground noise. Ground noise temperature in most situations is well approximated for soils by the value of 290 K, but is much less for surfaces that are highly reflective due to reflection of low temperature sky noise. Also, smooth sur­faces have high reflection for near grazing incidence angles.

Unlike ground noise, sky noise is a strong function of fre­quency. Sky noise is made up of atmospheric, cosmic, and manmade noise. Atmospheric noise increases with decreasing frequency below 1 GHz and is primarily due to lightning, which propagates over large distances via ionospheric reflec­tion below several MHz. Atmospheric noise increases with frequency above 10 GHz due to water vapor and hydrometeor absorption; these depend on time, season, and location. It also increases with decreasing elevation angle. Atmospheric gases have strong, broad spectral lines, such as water vapor and oxygen lines at 22 and 60 GHz, respectively.

Cosmic noise originates from discrete sources such as the sun, moon, and radio stars as well as our galaxy, which has strong emissions for directions toward the galactic center. Ga­lactic noise increases with decreasing frequency below 1 GHz. Manmade noise is produced by power lines, electric motors, and other sources and usually can be ignored except in urban areas at low frequencies. Sky noise is very low for frequencies between 1 and 10 GHz, and can be as low as a few K for high elevation angles.

Of course, the antenna pattern strongly influences antenna temperature; see Eq. (119). The ground noise temperature contribution to antenna noise can be very low for high-gain antennas having low side lobes in the direction of the earth. Broad beam antennas, on the other hand, pick up a signifi­cant amount of ground noise as well as sky noise. Losses on the antenna structure also contribute to antenna noise. A figure of merit used with satellite earth terminals is G/Tsys, which is the antenna gain divided by system noise tempera­ture usually expressed in dB/K. It is desired to have high values of G to increase signal and to have low values of Tsys to decrease noise, giving high values of G/Tsys.

Example: Direct Broadcast Satellite Reception. Reception of high quality television channels at home in the 1990s, with inexpensive, small terminals, is the result of three decades

 0.46 у ~2~

 n _ 4?r a 4?r Gr — — eapAp — (0 024)2

 0.7n

 2538

— 34 dB (70% aperture efficiency)

The received power from Eq. (114) is

PT(dBm) — 20.8 + 34.2 + 34 — 20 log(38, 000) — 20 log(12450)

— 32.44 — —116.9dBW (121)

This is 2 X 10~12 W! Without the high gains of the antennas (68 dB combined) this signal would be hopelessly lost in noise.

The receiver uses a 67 K noise temperature low noise block downconverter. This is the dominant receiver contribution, and when combined with antenna temperature leads to a sys­tem noise temperature of Tsys = 125 K. The noise power in the effective signal bandwidth Af = 20 MHz is

PN — kTsysAf

— 1.38 x 10-23 • 125 • 20 x 106 — 3.45 x 10-14 (122)

— —134.6dBW

Thus the carrier to noise ratio from Eqs. (118) and (121) is

CNR(dB) — PD(dBW) — PN(dBW)

 (123)

 — -116.9 — (-134.6) — 17.7 dB

 )

The bandwidth of narrow band antennas is usually expressed as a percent whereas wide band antennas are quoted as a ratio. Resonant antennas have small bandwidths. For exam-

ple, half-wave dipoles have bandwidths of up to 16%, (fU and fL determined by the VSWR = 2.0). On the other hand, anten­nas that have traveling waves on them rather than standing waves (as in resonant antennas), operate over wider fre­quency ranges.

Updated: 26.02.2014 — 15:00