Lecture 17: Antenna radiation -- near and far field, isotropic model/gain, Friis equation

Near vs. far fieldNear field: 
 |E| \propto \frac{1}{R^2} ∣E∣∝1R2 |E| \propto \frac{1}{R^2} ∣E∣∝R21​
, reactive -- can cause parasitic loading or near field coupling. The ratio of E to H is determined by the ratio of V to I and therefore the impedance of the circuit.
Kind of near field (where is it?), radiative -- spherical E and H waves.
Far field: 
 |E| \propto \frac{1}{R} ∣E∣∝1R |E| \propto \frac{1}{R} ∣E∣∝R1​
, radiative -- plane E and H waves. The ratio of E to H is  377 \: \Omega 377 Ω 377 \: \Omega 377Ω

(in free space).
Boundary between near and far field is 
 \text{max}\bigg(10D, 10\lambda, \frac{2D^2}{\lambda}\bigg) max(10D,10λ,2D2λ) \text{max}\bigg(10D, 10\lambda, \frac{2D^2}{\lambda}\bigg) max(10D,10λ,λ2D2​)
, where D is the length of the antenna from tip to tip. 
﻿
Isotropic model/radiation shapeThe "ideal" antenna model is isotropic, which is impossible in reality because it's impossible to have a continuous E field on a sphere where every point is tangent to the surface (hairy ball problem). But it's a common reference for antenna behavior.
The Equivalent Isotropic Power Density, 
 EIPD = \frac{P_{tx}}{4\pi r^2} EIPD=Ptx4πr2 EIPD = \frac{P_{tx}}{4\pi r^2} EIPD=4πr2Ptx​​
﻿
﻿
The behavior of an actual antenna is compared with an isotropic one using a unit called dBi:
dBi = 10 \log \bigg(\frac{P_{rad}(r,\theta,\phi)}{EIPD(r)}\bigg)
dBi=10log⁡(Prad(r,θ,ϕ)EIPD(r))dBi = 10 \log \bigg(\frac{P_{rad}(r,\theta,\phi)}{EIPD(r)}\bigg)dBi=10log(EIPD(r)Prad​(r,θ,ϕ)​)
The beam width of an antenna is the angle where the radiation power is above -3 dBi.
The max theoretical dBi (based on the shape of radiation) is called the "directivity," the measured max dBi is the "gain" (it takes into account loss). For a dipole antenna directivity is 1.76dBi, for a patch antenna it's ~3dBi.
﻿
Receiving antennasAntennas are reciprocal! All transmitting properties apply to receiving. This gives us an antenna model as an RLC circuit with a voltage source.
﻿
Receiving antenna behavior is specified by a parameter called gain, also comparing the received power to what would be received by an isotropic receiving antenna.
G = \frac{P_{rx}}{P_{iso}}
G=PrxPisoG = \frac{P_{rx}}{P_{iso}}G=Piso​Prx​​
There's also a thermodynamic derivation for something called the aperture of an antenna, an area proportional to how much power it receives. The effective aperture is the aperture in a specific direction for a non-isotropic antenna.
A_{iso} = \frac{\lambda^2}{4\pi}, A_\text{effective} = G \frac{\lambda^2}{4\pi}
Aiso=λ24π,Aeffective=Gλ24πA_{iso} = \frac{\lambda^2}{4\pi}, A_\text{effective} = G \frac{\lambda^2}{4\pi}Aiso​=4πλ2​,Aeffective​=G4πλ2​
Friis equationThe Friis equation puts everything together.
P_{rx} = P_{tx} \frac{1}{4\pi r^2} G_{tx} G_{rx} \frac{\lambda^2}{4 \pi} = P_{tx} G_{tx} G_{rx} \bigg(\frac{\lambda}{4 \pi r}\bigg)^2
Prx=Ptx14πr2GtxGrxλ24π=PtxGtxGrx(λ4πr)2P_{rx} = P_{tx} \frac{1}{4\pi r^2} G_{tx} G_{rx} \frac{\lambda^2}{4 \pi} = P_{tx} G_{tx} G_{rx} \bigg(\frac{\lambda}{4 \pi r}\bigg)^2Prx​=Ptx​4πr21​Gtx​Grx​4πλ2​=Ptx​Gtx​Grx​(4πrλ​)2
The quantity 
 (\frac{\lambda}{4 \pi r})^2 (λ4πr)2 (\frac{\lambda}{4 \pi r})^2 (4πrλ​)2
 is the "path loss", the loss that occurs regardless of the gain of the receiving and transmitting antennas.
The log version of the equation:
P_{rx} = P_{tx} + G_{tx} + G_{rx} - 20 \log \bigg( \frac{\lambda}{4 \pi r} \bigg)
Prx=Ptx+Gtx+Grx−20log⁡(λ4πr)P_{rx} = P_{tx} + G_{tx} + G_{rx} - 20 \log \bigg( \frac{\lambda}{4 \pi r} \bigg)Prx​=Ptx​+Gtx​+Grx​−20log(4πrλ​)
Lecture 17: Antenna radiation -- near and far field, isotropic model/gain, Friis equation

Near vs. far field

Isotropic model/radiation shape

Receiving antennas

Friis equation

Comments (loading...)

e157: radio frequency circuit design
