Fundamentals of Antennas Carlos A. Fernandes

Antennas and Propagation - Master in Aerospace Engineering

Carlos A. Fernandes

Fundamentals of Antennas

ANTENNAS AND PROPAGATION - MAero [email protected] 2

Source Transmission structure Antenna Radiation

4.1 Introduction to antennas

𝒁𝟎𝒁𝑻𝑬


Source Transmission structure Antenna Radiation




Most visible type of antennas


Half-wavelength dipole radiation

l / 2

4.4 Antenna radiation


Can be viewed as a section of a largerantenna

jyx

z

r

Rq

iq

ijir

dldl < l

4.2 Hertz dipole

ഥ𝐀 =𝜇

4𝜋ҧ𝐉 𝑑𝑉

𝑒−𝑗𝑘 𝑅

𝑅

Infinitesimal current


Radiation fields

4.2 Hertz dipole

2𝜋 Τ𝑅 𝜆 sphericalwave


Radiation fields

4.2 Hertz dipole

jy

x

z

r

Rq

iq

ijir

dl

Infinitesimal current

sphericalwave


4.3 Radiation pattern

= ℜ 𝑺 + 𝑗 ℑ 𝑺 (𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑐𝑎𝑠𝑒)

𝑈 = 𝑟2 𝑺

z

x

y

j

3𝐷 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑈/𝑈𝑚𝑎𝑥∝ (sin 𝜃)𝟐

(𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑓 𝜑 𝑓𝑜𝑟 𝐻. 𝐷. )


Polar representation


𝐸 − 𝑝𝑙𝑎𝑛𝑒 𝐻 − 𝑝𝑙𝑎𝑛𝑒


50 40 30 20 10 0

(u1)

(u2) (u3)AL360P AL560

AL760

Anechoic chamber – roll over azimuth configuration

(g2) (g3)

(g1)

RXTX

Controller

PC


Antena under test


Receive antenna

Antenna under test

Main lobe

Secondary lobes



Anechoic chamber – roll over azimuth configuration



Cartesian format




Types of radiation pattern


Rectangular horn (directive pattern)



4.3 Radiation pattern characterization


4.4 Directivity and Gain

~𝑃𝑖

𝑃𝑟

Geometry



Directivity of Hertz dipole

,

නsin3𝜃 = −3

4cos( 𝜃) +

1

12cos(3𝜃)

𝑈(𝜃)

→ 𝐷 = 1.5 @ 𝜃 =𝜋

2(1.76 dBi)



Directivity estimation𝜃1

𝑃𝑟 ≈ 𝑈𝑚න

0

2𝜋

න

0

𝜃1

sin 𝜃 𝑑𝜃 𝑑𝜑 = 2𝜋𝑈𝑚 −cos𝜃 0𝜃1

= 2𝜋𝑈𝑚 −cos𝜃1 + 1 𝑆𝑒 𝜃 ≪ 1, cos𝜃 ≈ 1 −𝜃2

2

= 𝜋 𝑈𝑚 𝜃12 =

𝜋

4𝑈𝑚 𝜃3𝑑𝐵

2 ≈ 𝑈𝑚 𝜃3𝑑𝐵2

(𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦)

𝐷 =4𝜋 𝑈𝑚

𝑈𝑚 𝜃3𝑑𝐵2


4.4 Directivity and Gain𝐷(𝜃, 𝜑) =

𝑈(𝜃, 𝜑)

𝑃𝑟/4𝜋

~𝑃𝑖

𝑃𝑟

Gain

𝐺(𝜃, 𝜑) =𝑈(𝜃, 𝜑)

𝑃𝑖/4𝜋

[dBi]

𝐺𝑑𝐵 = 10 log𝐺

𝐺𝑑𝑖𝑝[dBd]

Τλ 2 dipole

𝐺𝑑𝑖𝑝 = 1.64

Γi = 0=𝑃𝑟𝑃𝑖


1 medida

2 medida

~

AETSonda

PP

Sonda

P~P

Cornetapadrao

G [dBi]

f [GHz]

Aspecto tipico da curva de calibracao

de uma corneta padrao

Gain comparison method

𝐺𝑑𝐵 = 𝐺𝑝𝑑𝐵 + 𝑃1𝑑𝐵𝑚 − 𝑃2𝑑𝐵𝑚

𝐺

𝐺𝑝


𝑃𝑖 𝑃1

𝑃2𝑃𝑖

1𝑠𝑡 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡

2𝑛𝑑 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡

𝐴𝑈𝑇𝑝𝑟𝑜𝑏𝑒

𝑝𝑟𝑜𝑏𝑒𝑆𝑡𝑑 𝑔𝑎𝑖𝑛ℎ𝑜𝑟𝑛

𝑃1 ∝ 𝐺 𝑃𝑖

𝑃2 ∝ 𝐺𝑝 𝑃𝑖


4.5 Input impedance

𝑍𝐴 = 𝑅𝑟 + 𝑅𝐿 + 𝑗 𝑋𝐴

~𝑅𝐴

𝑋𝐴

𝑃𝑟

Application to the Hertz dipole


Hertz dipole l/2 dipole

4.5 Input impedance

𝑍𝐴 = 𝑅𝑟 + 𝑅𝐿 + 𝑗 𝑋𝐴 = 𝑅𝐴 + 𝑗 𝑋𝐴

= 0 in the resonance

𝐹𝑟𝑒𝑞 [GHz] 𝐹𝑟𝑒𝑞 [GHz]


4.5 Input impedance (l/2 dipole)

|𝛤|

1.5 GHz

2.2 GHz

𝑩~𝟏𝟎%


Polarization elipse

4.6 Polarization

iq

ij

ℰ(t)

b

a g

Eq

Ejleft-hand

right-hand

linear,

circular


Polarization mismatch (coordinate system)

4.6 Polarization

𝑐𝑝 = ത𝐞i ∙ ത𝐞a∗ 2 = cos ҧ𝜉

2Ƹ𝐢𝜃

Ƹ𝐢𝜑

ො𝐞a

ො𝐞i𝜉


q

180 90 0 90 180

180 90 0 90 180

4.7 Phase center

d

Eixo de rotação

Posição do centro de fase


4.7 Phase center


4.8 Band width

Gain, SLL

Input impedance

Polarization

Efficiency


Input reflection of l/2 dipole vs frequency

4.8 Band width

Reflection coefficient – magnitude and phase

𝐿𝐵 =0.15 GHz

1.8 GHz= 8.33%


Input reflection of a UWB antenna

4.8 Band width

Reflection coefficient – magnitude

𝐿𝐵 3.5 ∶ 1

10.5 𝐺𝐻𝑧

3 GHz


Received power, Friis formula

4.8 Effective aperture

𝑆𝑖 𝐴

[W/m2] [m2]

𝑒𝑀

𝑍𝐿Antenna

Load

𝑃𝑅 =1

2𝑅𝐿 𝐼𝐿

2 =𝑉𝐿

2𝑅𝐿2 𝑍𝐴 + 𝑍𝐿

2=

𝑉𝐿2

8 𝑅𝑟𝑐𝑜𝑚 𝑍𝐿 = 𝑍𝐴

∗

𝑃𝑅 =

ഥ𝑬 𝟐

2 𝑍0




𝑍𝐿Antenna

Load𝑃𝑅 = 𝑆𝑖𝐴𝑒𝑀 𝑃𝑅 =

𝑉𝐿2

8 𝑅𝑟𝐴𝑒𝑀 =

1

𝑆𝑖

𝑉𝐿2

8 𝑅𝑟

Application to the Hertz dipole

𝑑ℓ ≪ ℓ

𝑉 = ത𝐸 𝑑ℓ

=ഥ𝑬 2

2 𝑍0𝐴𝑒𝑀 =

3𝜆2

8 𝜋




𝑍𝐿Antenna

Load

𝑑ℓ ≪ ℓ

𝐴𝑒𝑀 =3𝜆2

8 𝜋

𝐷 = 𝑐𝑘 𝐴𝑒𝑀

3

2= 𝑐𝑘

3𝜆2

8 𝜋𝑐𝑘 =

4𝜋

𝜆2𝐷 =

4𝜋

𝜆2𝐴𝑒𝑀

𝜃2𝜃1 =𝜆2

𝐴𝑒𝑀

Universal constant

Use Hertz dipole results to obtain 𝑐𝑘

𝐷 = 1.5




𝐺𝑇(𝜃𝑇, 𝜑𝑇)

𝑟 𝐺𝑅(𝜃𝑅 , 𝜑𝑅)

ReceiverMatchingcircuit

Matchingcircuit

Free space loss

𝑆𝑖 𝑟 =𝑃𝑖

4𝜋𝑟2[𝑊/𝑚2]𝐺𝑇 𝜃𝑇 , 𝜑𝑇

𝑃𝑅 𝑟 =𝑃𝑖

4𝜋𝑟2[𝑊]𝐺𝑇 𝜃𝑇, 𝜑𝑇 𝐴𝑒𝑅(𝜃𝑅, 𝜑𝑅)

𝑃𝑅(𝑟) = 𝑃𝑖 𝐺𝑇 𝜃𝑇 , 𝜑𝑇 𝐺𝑅 𝜃𝑅 , 𝜑𝑅

𝜆

4 𝜋 𝑟

2

[𝑊]𝑐𝑝 𝑐𝑖 𝑐𝑚




𝑟

𝐺𝑇(𝜃𝑇, 𝜑𝑇)𝐺𝑅(𝜃𝑅 , 𝜑𝑅)

ReceiverMatchingcircuit

Matchingcircuit

𝑃𝑅 = 𝑃𝑖 𝐺𝑇 𝜃𝑇 , 𝜑𝑇 𝐺𝑅 𝜃𝑅, 𝜑𝑅

𝜆

4 𝜋 𝑟

2

𝑐𝑝 𝑐𝑖 𝑐𝑚


What is the limit on the received power (ex communications)

𝑃𝑖 = 43 𝑑𝐵𝑚, 𝐺𝑇 = 20 𝑑𝐵𝑖, 𝐺𝑅 = 73 𝑑𝐵𝑖, 𝑑 = 280 Mkm, 𝑓 = 8 GHz

𝑃𝑅 = −143. 5 𝑑𝐵𝑚 (down-link usually not done directly )

4.9 Thermal noise


4.9 Thermal noise


4.9 Thermal noiseEffect on received power (ex radiation pattern measurement)


Noise in a resistor

Electrons in any resistor 𝑅 at physical temperature 𝑇 ≠ 0 exhibit random

motion, responsible for a fluctuating resistance, and associated fluctuating

voltage across the resistor terminals 𝑉 = 0, 𝑉𝑟𝑚𝑠 = 𝑉2 ≠ 0. It exists

even if a current is not flowing in the resistor.

𝑅

𝑇

Physical resistor in open circuit

𝑉𝑟𝑚𝑠

𝐾 Boltzman constant, 1.38 × 10−23 [J/K]

𝑇 Absolute physical temperature [K]

∆𝑓 Bandwidth [Hz]

Ex: 𝑇 = 300 𝐾, 𝑅 = 50 𝛺, ∆𝑓 = 10 𝑀𝐻𝑧 → 𝑉𝑟𝑚𝑠 = 2.88 𝜇𝑉

𝑉𝑟𝑚𝑠 = 4 𝐾 𝑇 𝑅 ∆𝑓 [V]

Nyquist and Johnson (researchers from Bell

Labs) showed in 1928 that, in open circuit,

the root mean square of this voltage is

4.9 Thermal noise


the noise power transferred to the load is

4.9 Thermal noiseNoise in a resistor

𝑁 =𝑉𝑟𝑚𝑠2

4 𝑅

𝑅

𝑅𝑁

Equivalent circuit

𝑅

𝑇

Physical resistor

“Ideal noiseless”

resistors

Resistor 𝑅 connected to a load with the same value 𝑅

The transferred noise power is independent of 𝑅 value (as long

as the load has the same value as the resistor).

𝑉𝑟𝑚𝑠

𝑉𝑟𝑚𝑠

Ex: 𝑇 = 300 𝐾, ∆𝑓 = 10 𝑀𝐻𝑧 → 𝑁 = 0.0414 𝑝𝑊 (−103 𝑑𝐵𝑚)

𝑉𝑟𝑚𝑠 = 4 𝐾 𝑇 𝑅 ∆𝑓

𝑇 = 150 𝐾 → 𝑁 = 0.0414 𝑝𝑊 (−106 𝑑𝐵𝑚)

= 𝐾 𝑇 ∆𝑓

4.9 Thermal noiseEquivalent noise temperature

Very small noise power values are impractical to handle in calculations. It is easier to introduce the concept of equivalent noise temperature 𝑇𝑒 even if the loss mechanism is not purely thermal, and work only with these temperatures instead.

𝑇𝑒 =𝑁

𝐾 ∆𝑓


4.10 Antenna noise temperatureInternal noise temperature

Antennas contribute with two noise terms:

𝑁𝐴𝑖 associated with its loss resistance 𝑅𝐿𝑁𝐴𝑒 associated with external noise captured through the radiation

pattern.

𝑃𝑅 = 𝜂 𝑃𝑖𝑛 =𝑅𝑟𝑎𝑑

𝑅𝑟𝑎𝑑 + 𝑅𝐿𝑃𝑖𝑛

𝑁𝐴𝑖 = 𝐾 𝑇𝐴 1 − 𝜂 𝛥𝑓

𝑃𝑖𝑛 = 𝑆𝑖 𝐴𝑒𝑀 𝑃𝑅

𝑁𝐴 = 𝑁𝐴𝑖 + 𝑁𝐴𝑒

𝑇𝐴𝑖 = 𝑇𝐴 1 − 𝜂

𝐾 𝛥𝑓 𝑇𝐴 = 𝐾 𝛥𝑓( 𝑇𝐴𝑖 + 𝑇𝐴𝑒) 𝑇𝐴 = 𝑇𝐴𝑖 + 𝑇𝐴𝑒


4.10 Antenna noise temperatureExternal noise temperature

𝑇𝐴𝑒 =1

4𝜋න

0

2𝜋

න

0

𝜋

𝑇𝐵 𝜃, 𝜑 𝐺 𝜃, 𝜑 sin 𝜃 𝑑𝜃 𝑑𝜑

𝐺 𝜃, 𝜑 - Antenna radiation pattern

External noise sources associated with 𝑁𝐴𝑒:

• Man-made noise;

• Atmospheric noise;

• Galactic noise;

• Cosmic background.

𝑁𝐴𝑒 = 𝐾 𝑇𝐴𝑒 ∆𝑓

𝑇𝐵 𝜃, 𝜑 = 𝜖 𝑇𝑝 is the brightness temperature of the source, where 𝑇𝑝is the physical temperature of the source and its emissivity is 0 < 𝜖 < 1


4.14 Noise temperatureBrightness temperature


A – Quiet Sun

B – Moon

C – Galactic noise

D – Cosmic background

4.14 Noise temperatureBrightness temperature


Antenna noise temperatureExternal noise temperature

𝑇𝐴𝑒 ≈ 𝑇𝐵 𝜃0, 𝜑01

4𝜋න

0

2𝜋

න

0

𝜋

𝐺 𝜃, 𝜑 sin 𝜃 𝑑𝜃 𝑑𝜑

External noise sources associated with 𝑁𝐴𝑒:

In the specific case of a high directivity antenna, when its beamwidthcaptures only a small portion of an extended noise source, so that 𝑇𝐵 𝜃, 𝜑 can be considered constant within the antenna main beam, we can write:

= 1𝑇𝐴𝑒 ≈ 𝑇𝐵 𝜃0, 𝜑0

Extendednoise source

Fundamentals of Antennas Carlos A. Fernandes

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