# Chapter 9 Summary

# Concepts

- An antenna is a transducer between a guided wave propagating on a transmission line and an EM wave propagating in an unbounded medium, or vice versa.

- Except for some solid-state antennas composed of non-linear semiconductors or ferrite materials, antennas are reciprocal devices; they exhibit the same radiation patterns for transmission as for reception.

- In the far-field region of an antenna, the radiated energy is approximately a plane wave.

- The electric field radiated by current antennas, such as wires, is equal to the sum of the electric fields radiated by all the Hertzian dipoles making up the antenna.

- The radiation resistance R rad of a half-wave dipole is 73 Ω, which can be easily matched to a transmission line.

- The directional properties of an antenna are described by its radiation pattern, directivity, pattern solid angle, and half-power beamwidth.

- The Friis transmission formula relates the power received by an antenna to that transmitted by another antenna at a specified distance away.

- The far-zone electric field radiated by a large aperture (measured in wavelengths) is related to the field distribution across the aperture by Kirchhoff’s scalar diffraction theory. A uniform aperture distribution produces a far-field pattern with the narrowest possible beamwidth.

- By controlling the amplitudes and phases of the individual elements of an antenna array, it is possible to shape the antenna pattern and to steer the direction of the beam electronically.

- The pattern of an array of identical elements is equal to the product of the array factor and the antenna pattern of an individual antenna element.

# Mathematical and Physical Models

Antenna Properties

Pattern solid angle Ω p=∬4 π F(θ,ϕ)d Ω

Directivity D=4 π Ω p

Gain G=ξ D, ξ=P rad P rad+P loss

Effective area A e=λ 2 D 4 π

Far-field distance R>2 d 2 λ

Short Dipole (l≪λ)

E˜θ=j I 0 l k η 0 4 π e−j k R R sin θ

H˜ϕ=E˜θ η 0

S(R,θ)=η 0 k 2 I 0 2 l 2 32 π 2 R 2 sin 2 θ

D=1.5

β=90°

R rad=80 π 2(l/λ)2

λ/2 Dipole

E˜θ=j 60 I 0 cos[(π/2) cos θ]sin θ e−j k R R

H˜ϕ=E˜θ η 0

S(R,θ)=15 I 0 2 π R 2 cos 2[(π/2) cos θ]sin 2 θ

D=1.64

β=78°

R rad≈73 Ω

Friis Transmission Formula

P rec P t=G t G r λ 4 π R 2 F t(θ t,ϕ t)F r(θ r,ϕ r)

Antenna Arrays

Multiplication Principle

S(R 0,θ,ϕ)=S e(R 0,θ,ϕ)F a(θ)

Uniform Phase

F a(γ)=∑i=0 N−1 a i e j i γ 2, with γ=k d cos θ=2 π d λ cos θ

Linear Phase

F a(θ)=∑i=0 N−1 a i e j i γ′2, with γ′=k d cos θ−δ

Rectangular Aperture (Uniform)

S(R,θ)=S 0 sinc 2(π l x sin θ/λ),x–z plane

S(R,θ)=S 0 sinc 2(π l y sin θ/λ),y–z plane

β x z=0.88 λ l x, β y z=0.88 λ l y

D=4 π A e λ 2≈4 π A p λ 2

# Important Terms

Provide definitions or explain the meaning of the following terms:

- 3 dB beamwidth

- antenna

- antenna array

- antenna directivity D

- antenna gain G

- antenna input impedance

- antenna pattern

- antenna polarization

- aperture distribution

- array distribution

- array factor F a(θ,ϕ)

- azimuth angle

- beamwidth β

- broadside direction

- effective area (effective aperture) A e

- electronic steering

- elevation and azimuth planes

- elevation angle

- end-fire direction

- far-field (or far-zone) region

- feeding coefficient

- frequency scanning

- Friis transmission formula

- half-power beamwidth

- isotropic antenna

- linear phase distribution

- loss resistance R loss

- null beamwidth

- pattern multiplication principle

- pattern solid angle Ω p

- power density S(R,θ,ϕ)

- Poynting vector

- principal planes

- radiation efficiency ξ

- radiation intensity (normalized) F(θ,ϕ)

- radiation lobes

- radiation pattern

- radiation resistance R rad

- reciprocal

- scan angle

- short dipole (Hertzian dipole)

- signal-to-noise ratio S n

- solid angle

- spherical propagation factor

- steradian

- system noise temperature T sys

- tapered aperture distribution

- zenith angle