linear antennas

Linear AntennasRadiation Group

Signals, Systems and Radiocommunications Department

Universidad Politécnica de MadridAndrés García Aguilar, Pablo padilla de la Torre

E-Mail: andreg@gr ssr upm es ppadilla00@gr ssr upm es

ANTENNA DESIGN AND MEASUREMENT TECHNIQUES - Madrid (UPM) – March 2008

E Mail: [email protected], [email protected]

Linear Antennas

Outline

1 I t d ti1.- Introduction

2.- The electric dipole, Baluns

3.- Monopole over ground plane

4.- Dipoles Parallel to the ground plane

5.- Yagi-Uda Antennas

6.- Other linear antennas


Linear Antennas

• Linear antennas are considered to be those antennas that imply the use of electrically thin conductors (wavelength >>conductor diameter).

• Electric current flows over the conductor surface.

• In order to calculate radiated fields in these antennas, conductors are modeled as if they were current lines with no diameter. Its equivalent

i h f f h l d lcurrent is the surface current of the real model.


Linear Antennas

• Paying attention to the geometry, it is necessary to calculate the potential vector

Potential Vector:

( ) ( )∫ ′

′−−

′′−

′=

L

rrjk

ldrr

erIrA

0

40

πμ

Potential Vector :( )rr <<′

( )∫ ′− jkre ˆμ ( )∫ ′

′⋅ ′′=L

rrjk lderIr

eA ˆ

4πμ

• Potential vector allows us to obtain the electric field distribution:

AH ×∇=1

μH

jE ×∇=

1

ωε


0μ j 0ωε

Linear Antennas

1.- Introduction

2.- The electric dipole, Balunsp ,


4.- Dipoles Parallel to the ground planepo es a a e o e g ou d p a e


6.- Other linear antennas6. Other linear antennas


The electric dipole2

L

• Mainly, an electric dipoles formed by two wires working together.2

L

z

I(z)

Although it is possible to calculate the analyticalcurrent distribution over the structure it isI(z)

L

θIINv

current distribution over the structure, it ispreferable to assume some approximate resultsusing Transmission line theory

I(z)

vOpen circuit

L/2v

In that conditions, we obtain the current distribution

I z I sin kL

z zL

m( ) = −⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ <

2 2


The electric dipole

L L⎛⎜

⎞⎟

⎡ ⎤ L⎡ ⎤Feeding current:

II z I sin kL

z zL

m( ) = −⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ <

2 2 I I sin kL

IN m= ⎡⎣⎢

⎤⎦⎥2

⎥⎦⎤

⎢⎣⎡

=

2L

ksin

II IN

m

• The model yields a quite good agreement with the analytical result when considering λ

=L

L/22

=L

Current distribution for different dipole lengths: MoM solutionz

zI

z

Current distribution for different dipole lengths: MoM solution

Short dipole: Hertz dipole:

z

Im

I(z)Im

I(z)Im I(z)


L=λ/2 L=λL<λ/2IIN=Im

IIN=0

The electric dipole

L L

Current distributions calculated using Moment Method


Radiated Field

( ) ′−−′ rrjk

Infinitesimal Dipole

Potential Vector: ( ) ( )∫ ′

−−

′′−

′

π

μ=

L

rrjk0 ld

rr

erI

4rA

0

z

( )J r′

1

A1

H0

×∇μ

=′r

Pj

r

( )J rr r− ′

'r

dV

Hj

1E

0

×∇ωε

=

Electric and magnetic fields: ( ) rjkr ejk

IdlsenArAH 0

0

1

4ˆˆ −⎟

⎠⎞

⎜⎝⎛ +=⎥⎦

⎤⎢⎣⎡ −=

θφ∂θ∂

∂∂φ θ

x y

rjkerr

jk

r

ksen

rr

jkr

k

IdljE

rrr

0

320

20

320

0

1

2ˆ1

cosˆ2

4

−⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++−+⎟

⎠⎞

⎜⎝⎛ +=

⎠⎝⎥⎦⎢⎣θθθ

πη

π∂θ∂


Radiated Field

Finite Dipole'1'1 23

22

⎟⎞

⎜⎛

⎟⎞

⎜⎛ zz

potential vector:( )

∫′−⋅−

′′ rrjk

lderI

Aμ

Coordinates: ...sincos2

1sin

2

1cos'ˆ 2

22 +⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−=′− θθθθ z

r

z

rzrrr

( )

∫

∫

+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+−

⎞⎛⎞⎛

′−

′⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

=

=′=

2/ ...sincos2

'1sin

2

'1cos'

'1'1ˆ

2

4

4

23

22

2

32

Lz

r

z

rzjkrm

rr

zdze

zL

ksenI

ldA

θθθθμ

πμ

z

z’ I(z’)r̂

∫− +⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+−2/ ...sincos

2

'1sin

2

'1cos'4 2

3

22

2L z

r

z

rzr θθθθπz

L

θ

θ̂

⎟⎞

⎜⎛ 11

Electric field: ⎟⎟⎠

⎞⎜⎜⎝

⎛++∝ ....

11·

232

1 rkrkKE


Radiated Field

( ) ( )sen cos sen sen cos cosr r x y z z z z′ + + ′ ′θ φ θ φ θ θC di

Finite Dipole

Far potential vector:

( ) ( )sen cos sen sen cos cosr r x y z z z z⋅ ′ = + + ⋅ ′ = ′θ φ θ φ θ θCoordinates:

( ) =′′=′

′⋅−

∫ lderIr

eA

L

rrjkjkr

4ˆ

πμ

z

z’ I(z’)r̂

⎞⎛⎞⎛

=′⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −=

−

′−

∫kLkL

zdzezL

ksenIr

e L

L

zjkm

jkr

ˆ24

2/

2/

cos

πμ θ

z

L

θ

θ̂ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

=−

zsenr

sen

kLkL

k

I

r

e mjkr

ˆ

ˆˆcos2

coscos2

cos2

4 2θθθ

θ

θ

πμ

kL kLjkr

⎛⎝⎜

⎞⎠⎟ − ⎛

⎝⎜⎞⎠⎟− θcos cos cos

2 2

Could be considered as a 1st order approximation

( )E j A A je

rI

jkr

m= − + =⎝ ⎠ ⎝ ⎠

ω θ φ ηπ θ

θθ φ sen22 2

Far field:

Eφ = 0 Linear Polarization throughout θ


φ Linear Polarization throughout θ

Radiated Power

θ coscoscos ⎟⎞

⎜⎛−⎟

⎞⎜⎛ kLkL

k

z

r̂ ( ) θθ

θ

πηφθω φθ

ˆsin

2coscos

2cos

2ˆˆ

⎟⎠

⎜⎝

⎟⎠

⎜⎝=+−=

−

Ir

ejAAjE m

jkr

z’

L

θ

I(z’)r

Lθ̂ ( ) ( )

φθθη

φθφθ

π

ddE

rdUPrad )sin(2

,,

0

2

2

4∫∫∫ =Ω=

2

0

2

I

PR rad

rad =

Hertz dipole: Short dipole:

zz 2

2⎟⎞

⎜⎛π l

IP2

2 hertzplIP ⎟

⎞⎜⎛πz

I(z)Im

I(z)Im 003

⎟⎠⎞

⎜⎝⎛=

λη IPrad 412 00

hertzrad

pIP =⎟

⎠⎞

⎜⎝⎛=

λη

2280 ⎟

⎠⎞

⎜⎝⎛=

λπ l

Rrad

2220 ⎟

⎠⎞

⎜⎝⎛=

λπ l

Rrad


⎠⎝ λ ⎠⎝ λ

Radiation Parameters

Normalized Field diagrams:⎞⎛

θ

⎟⎠⎞

⎜⎝⎛ θ

π

sen

cos2

cos ( )1

2

+ cos cos

sen

π θθ

Multilobe Diagram

L=0.5λ L=λ L=1.5λ

Directivity: D0=1,64 = 2,15 dBi D0=2,41 D0=2,17


Rradiation: Rrad=73 Ω Rrad=∞ Ω Rrad=99,5 Ω

Input Impedance

Input impedance: (ZIN=Re+jXe)

L/2a

ZIN(λ/2)=73+j42,5 Ω when a → 0

a=dipole radius

L/λResonance Condition

⎡ ⎤λ %L/2a L = −⎡⎣⎢

⎤⎦⎥

λ2

1100

%Resonance


L/λ

L/2a

Dipole Feeding - Baluns

– Baluns are devices that transform balanced current lines into unbalanced ones Its name comes from: “balun” = balanced to unbalancedones. Its name comes from: balun = balanced to unbalanced.

– Baluns allow us to feed symmetric structures in a balanced way. y y

For instance: dipoles, fed with asymmetric transmission lines, coaxial wires… Used to transport energy from circuitry to the antenna.

B l d li

+V/2

Bifilar

Shielded Bifilar

Balanced lines:

-V/2 Coplanar

Unbalanced lines:

εr

+V

Coaxial

Microstripεr

Unbalanced lines:


0 Stripline εr

Baluns

I(z)

I2-I3 I1I2-I3 I1

z I2-I3 I1I1 ≠ I2-I3

L

I I

L

I I

L

I I

L

I I

I3I2

I3I2

λ/4I3=0

I2 I1

λ/4C i l

I3=0

I2 I1

h≈λ/4I3=0

I2 I1

h≈λ/4I3=0

I2 I1

LíneaCoaxial

I1

I2

LíneaCoaxial

I1

I2

I2

SectionCoaxial

Shortcircuit

I2

SectionBifilar

Shortcircuit

I1=I2I1=I2 I1

I2

I1

I2

Non symmetric feeding Bazooka or Sleeve Balun Broken Balun


Baluns

Equivalent Circuit L

b a

Zc ZIN Z jZ khBALUN b= tgZb

I3=0

a b

h

b

c

h=λ/4Ground plane

h

For h=λ/4 => ZBALUM= ∞Coaxial Líne

Zc ≈0,46λ0

w

λ0/4

t

0

Coaxial


Linear Antennas

1.- Introduction







Image Theorem

z ≥0 Valid Results

Jρ

Jρ

z ≥0 Valid Results

dV

ρ

dV

ρ

hz

( )E z 0 0 ( )E z 0 0

Perfect Electric ConductordV

h

( )E zt = =0 0>< ( )E zt = =0 0

dV

Ji

ρ ρi = −ρ J J x J y J z⎧ = + +⎧⎪

Sources and Image Currents

ρρ ρi

x y z

i x y z

J J x J y J z

J J x J y J z= −⎧⎨⎩

+ += − − +

⎧⎨⎪

⎩⎪


Vertical Monopole over Ground PlaneGround Plane

zz

I

h I(z)I(z)

><V

IIN

2V

IIN2h

I

DU

Pdd

d

= 4πV IIN

DU

Pmm= 4π

d

( )U Um d= ≤ ≤⎧

⎨⎪

⎪

0 2θ π

( )

ZV

ZINdi l INM l= =2

21

Pm

( )P U d d P= =∫∫22

θ φ θ θ φππ

sen

⎩⎪ ( )≤ ≤2 θπ πUm

= 0

D Dl di l= 2 Z ZINmonopolo INdipolo=1

2

ZI

ZINdipolo INMonopolo2( )P U d d Pm m d== ∫∫200

θ φ θ θ φθφ

, sen

d ldld RR1

=


D Dmonopolo dipolo= 2 p p2dipoloradmonopolorad RR2

Vertical Monopoles

MW Monopole over ground. Monopole over ground plane i l t d ith t lli isimulated with metallic wires

Typical Radiation Pattern


Linear Antennas

1.- Introduction







Image Theorem: parallel dipoles over ground planeover ground plane

><

z z

Ih><Ih

Ih

Ih

• When h<<λ, the radiated field is quite small due to the destructive summation of direct and reflected fieldsummation of direct and reflected field.

• When h=λ/4 the radiated field is enhanced due to constructive summation in z direction.

• Considering finite ground planes (higher than λ x λ), the image theorem can be applied and undesired effects can be neglected.


Antenna mutual couplings

• When designing composed antenas, with several radiating elements, it is t t d t l li b t th l tnecessary to study mutual couplings between these elements.

– From both, the radiation point of view (feeding currents) and the circuit point of view (impedance), the global antenna behaves as a multiport

V

V

Z Z Z

Z Z Z

I

IN1 11 12 1 1⎡

⎢⎤⎥

⎡⎢

⎤⎥

⎡⎢

⎤⎥I1 IN

linear network.

V

V

Z Z Z

Z Z Z

I

IN

N

N N NN N

2 21 22 2

1 2

2

⎣

⎢⎢⎢⎢

⎦

⎥⎥⎥⎥

=

⎣

⎢⎢⎢⎢

⎦

⎥⎥⎥⎥

⋅

⎣

⎢⎢⎢⎢

⎦

⎥⎥⎥⎥

1

I2

V1VN

N

...N N N NN N1 2⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Active impedance of i radiating l

ZV

IZ

I

IZ Z

I

Iii

iij

j

ij

N

ii ijj

ij

N

= = = +∑ ∑1 1

V2

element: I I Ii ij iji j

= =≠

1 1

Easily related to generalized multipole analysis

2

31


Easily related to generalized multipole analysis

N

Mutual Impedances between DipolesDipoles

(z=y) (z=y) (z=y)

kL/2

Mutual impedance between two identical parallel dipoles, one in front of the other, at λ/2

distance


distance

Linear Antennas

1.- Introduction







Yagi Antennas

• These antennas are made of a lot of dipoles. Typically, only one of them is t ll f d ( ti d i i l t) th t h i d d t d texternally fed (active, driving element); the rest have induced currents due to

mutual couplings (director elements).

• Passive dipoles behave whether reflecting (reflectors) or guiding (director elements) radiated fields.

Driving element

Reflectors

DiDirector elements

A Yagi antenna shows a radiation Gain, approximated by:

where n is the number of driving dipoles)1log(10 ++= nGG Dipoleyagi


where n is the number of driving dipoles.

Yagi Antennas

• 2 elements Yagi Uda antenna

P i Di l l >l “R fl t ”

z

θI

• 2 elements Yagi-Uda antenna.

Passive Dipole: l2>l1 “Reflector”l2<l1 “Director”

2l2d

I1

I2

Active dipole2l1

( )E E EI

e Ejkd= + ≈ +⎛⎜

⎞⎟ ⋅21 cosθ θ φ

xd cosθ

( )E E EI

e ET d= + ≈ +⎝⎜

⎠⎟ ⋅1 2

1

1 ,θ φ

V Z I Z I1 11 1 12 2= + I Z2 12= ZV

ZZ1 12

2

- Circuital equations:


Z I Z I21 1 22 20 = + I Z1 22

= − ZI

ZZIN = = −

111

22

2 elements Yagi Antenna: PatternsPatterns

H Plane (XZ Plane ): Ed(θ,φ=0)=constant; F(θ,φ=0)=|1+I2/I1 exp(jkdcosθ)|( ) d( ,φ ) ; ( ,φ ) | 2 1 p(j )|

0,6

0 4

0,80,6

0 4

Director

0,4

0,2

Active

0,4

0,2

z

DirectorActive

-z

ActiveReflector

a/λ= 0.003

a/λ= 0.003

E Plane (YZ Plane): The radiation patter must include the dipole radiation pattern (like a donut)

z

y


pattern (like a donut).

Yagi antennas.

Typical dimensions:

Si l i d lSimulation model:


Yagi Examples

Diedric reflector Yagi

Usually, as active element, it is used abended dipole in order to increaseinput impedance and band width.

Double reflector Yagi

Horn dipoles Yagi


Linear Antennas

1.- Introduction







Loop Antennas

Aproximate current distribution:

Short line with shortcircuit = uniform current

Short Loop (in terms of λ): Large Loop (in terms of λ):

l<<λ

bl=λ/2

22 ≈= bC π NullMax

Long line with shortcircuit

NullMax

•multilobe pattern, high efficiency.


Helix

• The helix geometry depends on the next• The helix geometry depends on the next values:

– D= Helix diameter. d

– C= Perimeter of the helix cylinder= πD

– S= length of one turn= πD tanαα Inclination angle atan(S/C)

D

– α= Inclination angle= atan(S/C)

– L= one turn length

– N= Number of turns.

S

A

– A= Total length= NS

– d= Wire diameter

• Helix antennas are usually used in its axial radiation mode, when C is similar to λ.

C=πD

α

L


radiation mode, when C is similar to λ.

S

HelixAxial Radiation ModeAxial Radiation Mode

• Electrically big helix, with dimensions 3/4<C/λ<4/3 y α ≈ 12º-15º:

Progressive current wave through the helix:– Progressive current wave through the helix: I(l)=I0exp(-jkl)

– Wide band: fsup/finf=1,78

Ω≈λ

≈ 140C

140Rin

– Approximately real impedance value:

λ

– Circular polarization (left or right handed, the same as the wire turn)

⎞⎛2

I(l)– Quite directive radiation pattern. Secondary lobe

level: -9 dB.

– Directivity:


λ≈

λ⎟⎠⎞

⎜⎝⎛

λ≈

A15

NSC15D

2Directivity:

Helix examples


linear antennas

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