Antenna Theory and Design

Associate Professor: WANG Junjun 王珺珺 School of Electronic and Information Engineering, Beihang University [email protected] 13426405497 Chapter 2 Part II Antenna Basics Chapter 2 Part II Antenna Basics

▫ Antenna apertures ▫ Antenna effective height ▫ Friis transmission equation ▫ Radar range equation 1. Antenna apertures • The concept of aperture is most simply introduced by considering a receiving antenna, to describe the power capturing characteristics of the antenna when a wave impinges on it. • In a given direction aperture is defined as “the ratio of the available power at the terminals of a receiving antenna to the power flux density of a plane wave incident on the antenna from that direction, the wave being polarization-matched to the antenna. If the direction is not specified, the direction of maximum radiation intensity is implied.”

 where 2 Ae = effective area (effective aperture) (m ) PT = power delivered to the load (W) 2 Wi = power density of incident wave (W/m ) 1. Antenna apertures

• It seems obvious to optical astronomers that a parabolic dish antenna that is many wavelengths across, will have an aperture nearly equal to their physical area. However other antenna such as a Yagi and Collinear arrays my not look to be the same at first glance but they do achieve the same result using other means at radio frequencies. • 1. Antenna apertures

• The maximum effective aperture (Aem) of any antenna is related to its maximum directivity (D0) by

• If reflection, conduction-dielectric and polarization losses are also included 1. Antenna apertures

• The aperture efficiency ap of an antenna, is defined as the ratio of the maximum effective area Aem of the antenna to its physical area Ap.

• For aperture type antennas, such as waveguides, horns, and reflectors, the maximum effective area cannot exceed the physical area but it can equal it (Aem ≤Ap or 0 ≤  ap ≤ 1). • The maximum effective area of wire antennas is greater than the physical area (if taken as the area of a cross section of the wire when split lengthwise along its diameter). Thus the wire antenna can capture much more power than is intercepted by its physical size! 2 Antenna effective height • An antenna in the receiving mode, whether it is in the form of a wire, horn, aperture, array, dielectric rod, etc., is used to capture (collect) electromagnetic waves and to extract power from them. • The effective height/length of an antenna, whether it be a linear or an aperture antenna, is a quantity that is used to determine the voltage induced on the open-circuit terminals of the antenna when a wave impinges upon it.

Uniform plane wave incident upon dipole and aperture antennas. 2 Antenna effective height

• The effective length represents the antenna in its transmitting and receiving modes, and it is particularly useful in relating the open-circuit voltage Voc of receiving antennas. This relation can be expressed as

where Voc = open-circuit voltage at antenna terminals Ei = incident electric field le = vector effective length

Uniform plane wave incident upon dipole and aperture antennas. 2 Antenna effective height

• The effective height/length of a linearly polarized antenna receiving a plane wave in a given direction is defined as “the ratio of the magnitude of the open- circuit voltage developed at the terminals of the antenna to the magnitude of the electric-field strength in the direction of the antenna polarization. The antenna vector effective length is used to determine the polarization efficiency of the antenna.

Uniform plane wave incident upon dipole and aperture antennas. 2 Antenna effective height

• The effective length is the length of a thin straight conductor oriented perpendicular to the given direction and parallel to the antenna polarization, having a uniform current equal to that at the antenna terminals and producing the same far-field strength as the antenna in that direction 3 Friis transmission equation

• The Friis Transmission Equation relates the power received to the power transmitted between two antennas separated by a distance R > 2D2/λ, where D is the largest dimension of either antenna.

Geometrical orientation of transmitting and receiving antennas for Friis transmission equation • let us assume that the transmitting antenna is initially isotropic. If the input power at the terminals of the transmitting antenna is Pt , then its isotropic power density W0 at distance R from the antenna is

 where et is the radiation efficiency of the transmitting antenna. 3 Friis transmission equation

• For a nonisotropic transmitting antenna, the power density in the direction θt, φt can be written as

▫ where Gt (θt,φt ) is the gain and Dt (θt,φt ) is the directivity of the transmitting antenna in the direction θt, φt . • Since the effective area Ar of the receiving antenna is related to its efficiency er and directivity Dr by

• The amount of power Pr collected by the receiving antenna can be written as,

• or the ratio of the received to the input power as 3 Friis transmission equation

• The power received based on last equation assumes that the transmitting and receiving antennas are matched to their respective lines or loads and the polarization of the receiving antenna is polarization-matched to the impinging wave. 3 Friis transmission equation

• For reflection and polarization-matched antennas aligned for maximum directional radiation and reception,

▫ This is known as the Friis Transmission Equation. It relates the power Pr (delivered to the receiver load) to the input power of the transmitting antenna Pt . ▫ The term (λ/4πR)2 is called the free-space loss factor. And it takes into account the losses due to the spherical spreading of the energy by the antenna. 4 Radar range equation

• Let us assume that the transmitted power is incident upon a target, as shown below:

Geometrical arrangement of transmitter, target, and receiver for radar range equation.

• We now introduce a quantity known as the radar cross section or echo area (σ ) of a target which is defined as the area intercepting that amount of power which, when scattered isotropically, produces at the receiver a density which is equal to that scattered by the actual target. 4 Radar range equation

 where  σ = radar cross section or echo area (m2)  R = observation distance from target (m) 2  Wi = incident power density (W/m )  Ws = scattered power density (W/m2)  Ei (Es) = incident (scattered) electric field (V/m)  Hi (Hs) = incident (scattered) magnetic field (A/m) 4 Radar range equation

• Using the definition of radar cross section, the transmitted power incident upon the target is initially captured and then it is reradiated isotropically, insofar as the receiver is concerned. The amount of captured power Pc is obtained by multiplying the incident power density by the radar cross section σ, or

• The power captured by the target is reradiated isotropically, and the scattered power density can be written as

• The amount of power delivered to the receiver load is given by 4 Radar range equation

• It can be written as the ratio of the received power to the input power, or

• If these two factors are also included,

 where  ˆρw = polarization unit vector of the scattered waves  ˆρr = polarization unit vector of the receiving antenna 4 Radar range equation

• For polarization-matched antennas aligned for maximum directional radiation and reception,

• This is known as the Radar Range Equation. It relates the power Pr (delivered to the receiver load) to the input power Pt transmitted by an antenna, after it has been scattered by a target with a radar cross section (echo area) of σ. • Conclusions 1.The calculation of antenna effective height and area. 2.The Friis transmission equation and radar range equation.

• .