ECE422: Radio and Microwave WirelessSystems

Antenna Radiation Patterns

University of Toronto

Prof. Sean Victor Hum

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Antenna Radiation Pattern

“A mathematical functionor graphical representationof the radiation propertiesas a function of spacecoordinates”

Field strength,directivity, radiationintensity, power density,phase, etc.

Normally plotted in thefar-field of the antennaPlotted at a fixed distance(radius) r from antenna

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Field vs. Power Patterns

Field Pattern Power Pattern

HPBW = half power beamwidth

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Radiation Pattern Lobes and Beamwidths

3D pattern

2D cut (e.g. yz-plane)

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Radiation Pattern Types

Isotropic – equal radiationin all directions

Not physically realizablebut a useful theoreticalreference

Omnidirectional –possessing anon-directional pattern ina given plane

Useful for broadcastscenarios

Directional – having apreferred direction oftransmission/reception

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Principal Planes and Patterns

E-plane – plane containingthe electric-field vector anddirection of maximumradiationH-plane – plane containingthe magnetic-field vectorand direction of maximumradiation

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Radiation Pattern of a Dipole

3D pattern E-plane cut H-plane cut

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Field Regions

Near-field region (r < R1) –non-radiating [reactive]antenna fields dominateRadiating near-field region(R1 < r < R2) – radiatingfields dominate butangular distributiondepends on rFar-field (r > R2) – angularfield distribution does notdepend on r

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Example: Paraboloidal Reflector

Field measurements in thedifferent regions produceseemingly differentradiation patternsOnly in the far-field are thepatterns invariant ofobservation distance!

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Pattern Quantification: Solid Angle

Radians are a measure ofplane angle: 1 rad subtendsan arc of length r

C = 2πr

rads =Cr

=2πr

r= 2π

Steradians are a measure ofsolid angle: 1 sr subtends aspherical area of area r2

A = 4πr2

srs =Ar2 =

4πr2

r2 = 4π

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Pattern Quantification: Solid Angle

Radiation patterns aredescribed using aggregateparameters that quantifypowerKnowing the fieldsproduced by an antenna,power density (Poyntingvector) can be computedPower is computed bytaking the surface integralof power density

Pav =12

Re[E ×H∗]

Wrad =

S

Pav · n ds′

P0 = r P0 = rWrad

4πr2

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Pattern Quantification: Surface Integrals

Radiation integrals oftenuse spheres as the closedsurfaceRadius of sphere chosen tobe in the far fieldda = r2 sinθdθdφ [m2]

dΩ =dar2 = sinθdθdφ [sr]

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Radiation Intensity

Definition: power radiated per unit solid angle

U = r2Pav,r [W/sr]

Wrad =

Ω

UdΩ =

∫ 2π

0

∫ π

0U sinθdθdφ

In the far field,

U(θ, φ) =r2

2η|E(r, θ, φ)|2 ≈

r2

2η

[|Eθ(r, θ, φ)|2 + |Eφ(r, θ, φ)|2

]

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Directivity (D)

Definition: the ratio of the radiation intensity in a givendirection to the radiation intensity averaged over alldirections

D depends on θ, φGenerally maximum directivity is of most interest

D =UU0

=4πUWrad

Dmax ≡ D0 =Umax

U0=

4πUmax

Wrad

Can also be expressed in decibels with respect to isotropicantenna (dBi): 10 log10(D)

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Generalized Form of Directivity

Define a pattern function such that

U = B0F(θ, φ)

Then,

Prad = B0

∫ 2π

0

∫ π

0F(θ, φ) sinθdθdφ

D(θ, φ) = 4πF(θ, φ)∫ 2π

0

∫ π0 F(θ, φ) sinθdθdφ

D0 =4π[∫ 2π

0

∫ π0 F(θ, φ) sinθdθdφ

]/F(θ, φ)|max

≡4πΩA

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Beam Solid Angle

ΩA =1

F(θ, φ)|max

∫ 2π

0

∫ π

0F(θ, φ) sinθdθdφ

Solid angle through whichall the radiated powerwould flow if its radiationintensity were constant(= Umax) for all angleswithin ΩA

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Directivity Approximations

High-gain antennas oftenhave most power focusedin a tight beam withnegligible sidelobes; then:

ΩA ≈ θ1rθ2r

D0 =4πΩA≈

4πθ1rθ2r

≈41, 253θ1dθ2d

where θ1 and θ2 are thehalf-power beamwidthsintwo orthogonal planes(rads/degs)

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