Radiation Power and Polarisation

Steradian Ω

• Unit solid angle

• SI unit for 3D angular measurement

Spherical Coordinates - Area

Spherical Coordinates - Volume

Radiation Density 𝑺(𝒓, 𝜽, 𝝓) [𝑾 / 𝒎^𝟐]

• Power radiated per unit area at distance r

• Isotropic Source Example:

  • 𝑺_i(𝒓, 𝜽, 𝝓) = P_rad / 4𝝅r²

Radiation Intensity 𝑼(𝜽, 𝝓) [𝑾/Ω]

• Power radiated per unit solid angle

  • Independent of distance

• Isotropic Source Example

  • 𝑼_i(𝜽, 𝝓) = 𝑷_𝒓𝒂𝒅 / 4𝝅

D𝐢𝐫𝐞𝐜𝐭𝐢𝐯𝐢𝐭𝐲 𝑫(𝜽, 𝝓)

• How well an antenna “beams” energy in a given direction

• 𝑫(𝜽, 𝝓) = 𝑼(𝜽, 𝝓) / 𝑼_i(𝜽, 𝝓) = 4𝝅 𝑼(𝜽, 𝝓) / 𝑷_𝒓𝒂𝒅

• Maximum directivity usually quoted

  • 𝑫_𝒎𝒂𝒙 = 𝑺_𝒎𝒂𝒙 /𝑺_𝒊

• Units in dBi

G𝐚𝐢𝐧 𝑮(𝜽, 𝝓)

• Similar to directivity but includes losses

• 𝑮(𝜽, 𝝓) = 4𝝅 𝑼(𝜽, 𝝓) / 𝑷_𝒊𝒏

Radiation Efficiency

• 𝑮 = 𝜼𝑫

• 𝜼 = radiation efficiency = 𝑷_𝒓𝒂𝒅 / 𝑷_𝒊𝒏

• Typical values of 0.5 to 0.95

Effective Isotropic Radiated Power

EIRP = 𝑷_𝒊𝒏 𝑮

Relative Gain

In many cases antenna radiation patterns are displayed using relative gain

• Normalises peak gain to 0dB

• 𝑮′(𝜽, 𝝓) = 𝑮(𝜽, 𝝓) / 𝑮𝒎𝒂𝒙
  

Beamwidth Approximation

• Approximate relationship between gain and beamwidth

• Assuming a directional antenna with 3dB beamwidths given by βx and βy

• If all power is assumed to be within 3dB beamwidth then at distance R

  • 𝑃_𝑟𝑎𝑑 ≈ 𝑆_𝑚𝑎𝑥𝛽𝑥𝛽𝑦𝑅²

  • Since 𝑆_𝑚𝑎𝑥 = 𝑃_𝑟𝑎𝑑𝐺_𝑚𝑎𝑥 / 4𝜋𝑅²

  • 𝐺_𝑚𝑎𝑥 = 4𝜋 / 𝛽𝑥𝛽𝑦

Polarisation

• What direction the E field is in (Far Field parameter)

• Linear / Circular / Elliptical

Instantaneous Field E(z,t)

Inclination angle of the electric field vector

Linear Polarisation

• Tip of E(z,t) traces a straight line

• Simplest case when a_x or a_y = 0

• Can also happen when both x and y componenrs are non-zero but are in phsae

Circular Polarisation

Occurs when magnitudes of x and y components are equal and have a phase difference of δ =± 𝜋/2

Elliptical Polarisation

In practice it is difficult to generate perfect circular polarisation and so the tip of E vector traces an ellipse

To measure the ‘ellipticity’ the axial ratio is used:

AR = major axis / minor axis

1 ≤ AR ≤ ∞
Circular → Linear

Polarisation Diversity

Allows us to transmit 2 separate signals using the same frequency - polarisation diversity

• Tx(1) antenna is vertically polarised and Tx(2)
  antenna is horizontally polarised

• Signal from Tx(1) will be received by Rx(1) antenna
  but not Rx(2)

Cross Polarisation

• In practice, antennas do not have perfect polarisation

• A measure of this is cross-polarisation (XPOL) level – usually in dBs.

• Typical XPOL levels are -20 to -40dB below COPOL (co-polar = main polarisation direction)

Left Hand vs Right Hand Circular Polarisation

Transmitter system hardware can be designed to make the wave spin in either counter-clockwise (LHCP) or clockwise (RHCP)

The receiver antenna is built in a way to accept the correct polarisation and reject the other