lecture 31 wire antennas - cornell university 31 wire antennas in this lecture you will learn: •...

1

ECE 303 – Fall 2006 – Farhan Rana – Cornell University

Lecture 31

Wire Antennas

In this lecture you will learn:

• Generation of radiation by real wire antennas

Short dipole antennasHalf-wave dipole antennasThree-half-wave dipole antennasSmall wire loop antennas – magnetic dipole antennas


Center Fed Wire AntennaConsider the following wire antenna fed via a transmission line:

dZo

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ''4

ˆ

''4

ˆ

''ˆ

'4

ˆ''4

'

cos'2

2

'ˆ.ˆ

'ˆ'

dzezIer

z

dzezIr

zrA

dzezzr

zIzdverrrJrA

zkjd

d

rkjo

zzrrkjoff

zzrkjorrkjo

θπµ

πµ

πµ

πµ

∫=

∫≈

∫−

=∫∫∫−

=

−

−

−−∞

∞−

−−∞

∞−

−−

rr

rrr

rrrr rrr

( ) ( ) ( ) ( )yxzIzrJ δδˆ=rr

x

z

I

If one is interested in radiation far-fields only, then assume:

rdr

<<⎭⎬⎫

⎩⎨⎧

πλ

2,

Far-field approximation -also called the Fraunhofferapproximation

( ) ( ) ( ) ( ) ( )θφθφθ cosˆsinsinˆcossinˆˆ zyxr ++=Recall that:

2


Center Fed Short Dipole Wire Antenna - IShort dipole wire antenna fed via a transmission line:

dZo


x

z

I

πλ

2<<dShort dipole ⇒

Make an assumption for the current distribution on the antenna – a triangular distribution

z

I

d/2

- d/2

I(z)

( ) ( ) ( )

( )

rkjeffo

d

d

rkjo

zkjd

d

rkjoff

er

Idz

dzzIer

z

dzezIer

zrA

−

−

−

−

−

=

∫≈

∫=

πµ

πµ

πµ θ

4ˆ

''4

ˆ

''4

ˆ

2

2

cos'2

2

rr

πλ

2' <<< dzSince:

Where: ( )22

''2

2

dddIdzzIdI effd

deff =⇒=∫=

−

A Hertzian-dipole-like solution


Center Fed Short Dipole Wire Antenna - IIShort dipole wire antenna fed via a transmission line:

dZo


x

z

I

πλ


z

I

d/2

- d/2

I(z)

( ) ( ) rkjeffff e

rIdkjrH −= θπ

φ sin4

ˆrr

( ) ( ) rkjeffoff e

rIdkjrE −= θ

πηθ sin

4ˆrr

The radiation from a short dipole looks like that from a Hertzian dipole except that d is replaced by deff

( ) ( ) ( )

( )θπ

η 22

*

sin42

ˆ

Re21,

rIdkr

rHrEtrS

effo

ffff

=

×=rrrrrr

( ) ( )

2

2

0 0

2

12

sinˆ.,

effo

rad

Idk

ddrrtrSP

πη

φθθπ π

=

∫ ∫=rr

3


Center Fed Short Dipole Wire Antenna - IIIShort dipole wire antenna fed via a transmission line:

dZo


x

z

I

πλ


z

I

d/2

- d/2

I(z)

Antenna Gain:

For a short dipole the gain is:

( )( )

( )θπ

φθ 22 sin

23

4

ˆ.,, ==

rP

rtrSG

rad

rr

Antenna Radiation Pattern:

( ) ( ) ( )θφθφθ 2

maxsin,, ==

GGp

0

180

90

30

60

120

150

θ0

180

90

30

60

120

150

θ

( )0, =φθp


Center Fed Half-Wave Dipole Wire Antenna - IHalf-wave dipole antenna fed via a transmission line:

dZo


x

z

I

2λ

=dHalf-wave dipole ⇒

Make an assumption for the current distribution on the antenna – a sinusoidal distribution

z

d/2

- d/2

I(z)

( ) ( ) ( )

( ) ( )

( )

( )θ

θπ

πµ

πµ

πµ

θλ

λ

θ

2

cos'4

4

cos'2

2

sin

cos2

cos

2ˆ

''cos4

ˆ

''4

ˆ

⎟⎠⎞

⎜⎝⎛

=

∫≈

∫=

−

−

−

−

−

rkjo

zkjrkjo

zkjd

d

rkjoff

erkIz

dzezkIer

z

dzezIer

zrArr

( ) ( )zkIzI cos=

4


Center Fed Half-Wave Dipole Wire Antenna - IIHalf-wave dipole wire antenna fed via a transmission line:

dZo


x

z

I

2λ


z

d/2

- d/2

I(z)

( )( )

( )θ

θπ

πµ

2sin

cos2

cos

2ˆ

⎟⎠⎞

⎜⎝⎛

= − rkjoff e

rkIzrA

rr

( ) ( )zkIzI cos=

( )( )

( )θ

θπ

πηθ

sin

cos2

cos

2ˆ

⎟⎠⎞

⎜⎝⎛

= − rkjoff e

rIjrE

rr( )

( )

( )θ

θπ

πφ

sin

cos2

cos

2ˆ

⎟⎠⎞

⎜⎝⎛

= − rkjff e

rIjrH

rr

This implies:


Center Fed Half-Wave Dipole Wire Antenna - IIIHalf-wave dipole wire antenna fed via a transmission line:

dZo


x

z

I

2λ


This total power radiated is:

( ) ( )

( )

( )( )

2

2

0 0

22

22

2

0 0

2

422.1

sinsin

cos2

cos

22

sinˆ.,

I

ddrrI

ddrrtrSP

o

o

rad

πη

φθθθ

θπ

πη

φθθ

π π

π π

≈

∫ ∫⎟⎠⎞

⎜⎝⎛

=

∫ ∫=rr

( )( )

( )θ

θπ

φθ 2

2

sin

cos2

cos64.1,

⎟⎠⎞

⎜⎝⎛

≈G

( )( )

( )θ

θπ

φθ 2

2

sin

cos2

cos,

⎟⎠⎞

⎜⎝⎛

=p

0

180

90

30

60

120

150

( )0, =φθp

θ Hertzian dipole

Half-wave dipole

Ω≈= 7322I

PR radrad

5


Center Fed Three-Half-Wave Dipole Wire Antenna - I

dZo


x

z

I

Make an assumption for the current distribution on the antenna – a sinusoidal distribution

z

d/2

- d/2

I(z)( ) ( )zkIzI cos=

( ) ( ) ( )

( ) ( )

( )

( )θ

θπ

πµ

πµ

πµ

θλ

λ

θ

2

cos'43

43

cos'2

2

sin

cos2

3cos

2ˆ

''cos4

ˆ

''4

ˆ

⎟⎠⎞

⎜⎝⎛

−=

∫≈

∫=

−

−

−

−

−

rkjo

zkjrkjo

zkjd

d

rkjoff

erkIz

dzezkIer

z

dzezIer

zrArr

23λ=dThree-half-wave dipole ⇒


Center Fed Three-Half-Wave Dipole Wire Antenna - II

d x

z

I

23λ

=d

( )( )

( )θ

θπ

φθ 2

2

sin

cos2

3cos,

⎟⎠⎞

⎜⎝⎛

∝p

( )( )

( )θ

θπ

πηθ

sin

cos2

3cos

2ˆ

⎟⎠⎞

⎜⎝⎛

−= − rkjoff e

rIjrE

rr

( )( )

( )θ

θπ

πφ

sin

cos2

3cos

2ˆ

⎟⎠⎞

⎜⎝⎛

−= − rkjff e

rIjrH

rr

z

d/2

- d/2

I(z)( ) ( )zkIzI cos=

0

180

90

30

60

120

150

( )0, =φθp θ

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Buddipole TM

A 16 ft dipole for 1-50 MHz radio

A 1-5 GHz home-made dipole antenna for Wireless LAN with a co-axial SMA RF feed

Home Made Dipole Antennas


49.92 MHz incoherent scatter radar at the Peru ObservatoryThe radar has an array of 18,432 half-wave dipoles !!

Radars for Upper Atmosphere Research

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A short dipole antenna integrated with a low noise amplifier on a PC board for mobile receivers(4-8 GHz)

Radial stub tuners for impedance matching

Antennas for Mobile Consumer Products

A PCMCIA card antenna with two crossed short dipoles –shown with the cover removed (for 2-5 GHz)


Small Wire Loop Antenna – A Magnetic Dipole Radiator

x

y

a 'sdr

z

I

rr

'φ

Consider a small loop of wire carrying time-varying current:

'rr

( ) ( )

( ) ( ) ''ˆ4

''4

'

'.ˆ2

0

'

φφπµ

πµ

πadeI

rrA

dverrrJrA

rrrkjoff

rrkjo

r

rr

rr

rr

rrrr

−−

−−

∫=

∫∫∫−

=

Note that:

( ) ( )'cosˆ'sinˆ'ˆ φφφ yx +−=

( ) ( )'sinˆ'cosˆ' φφ ayaxr +=r

( ) ( ) ( ) ( ) ( )θφθφθ cosˆsinsinˆcossinˆˆ zyxr ++=

( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( )[ ] '

'cosˆ'sinˆ4

'sinsin'coscossin

2

0

φ

φφπ

µ

φφφφθ

π

de

yxerIarA

akj

rkjoff

+

− +−∫=rr

This gives:

( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( )[ ][ ] ''sinsin'coscossin1

'cosˆ'sinˆ4

2

0φφφφφθ

φφπ

µ π

dakj

yxerIarA rkjo

ff

++

+−∫≈ −rr

Small loop ⇒ a << λ/2π

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Small Wire Loop Antenna - II

x

y

a 'sdr

z

I

rr

'φ

'rr

Fields are proportional to the product of the current and the area of the loop

a << λ/2π( ) ( ) ( ) rkjo

ff eraIkjrA −≈ θ

ππµφ sin

4ˆ

2rr

( ) ( ) ( ) rkjoff e

raIkrE −≈⇒ θ

ππηφ sin

4ˆ

22rr

( ) ( ) ( ) rkjff e

raIkrH −−≈⇒ θ

ππθ sin

4ˆ

22rr

Total power radiated is: ( ) 2412

IakP orad

ηπ=

Radiation resistance is: ( )42 62ak

IPR orad

rad ηπ==

( ) ( ) ( )θφθφθ 2

maxsin,, ==

GGpRadiation pattern is:

0

180

90

30

60

120

150

θ0

180

90

30

60

120

150

θ( )0, =φθp


N-turn Small Wire Loop Antenna

x

yz

I

Consider a small loop of wire carrying time-varying current:

Fields are proportional to the product of the current and the area of the loop

a << λ/2π( ) ( ) ( ) rkjo

ff er

aINkjrA −≈ θπ

πµφ sin4

ˆ 2rr

( ) ( ) ( ) rkjoff e

raINkrE −≈⇒ θ

ππηφ sin

4ˆ

22rr

( ) ( ) ( ) rkjff e

raINkrH −−≈⇒ θ

ππθ sin

4ˆ

22rr

Total power radiated is: ( ) 2412

INakP orad

ηπ=

Radiation resistance is: ( )422 62

akNIPR orad ηπ==

( ) ( ) ( )θφθφθ 2

maxsin,, ==

GGpRadiation pattern is:

It is easier to obtain larger radiation resistances with small loop antennas (containing many turns) than with short dipole antennas of the same size

N-turns

a

9


Electric Dipole Radiators Vs Magnetic Dipole Radiators

a

I(t)

a << λ/2π

x

z

dq(t)

-q(t)I(t)

z

x

d << λ/2π

( ) ( ) ( ) rkjoff e

raIkrE −= θ

ππηφ sin

4ˆ

22rr

( ) ( ) ( ) rkjff e

raIkrH −−= θ

ππθ sin

4ˆ

22rr( ) ( ) rkjff e

rIdkjrH −= θ

πφ sin

4ˆrr

( ) ( ) rkjoff e

rIdkjrE −= θ

πηθ sin4

ˆrr

( ) ( ) ( ) ( )[ ]θθθεπ

sinˆcos2ˆ4

, 3 += rrdtqtrEo

nfrr

( ) ( ) ( ) ( )[ ]θθθππ sinˆcos2ˆ

4, 3

2+= r

ratItrHnf

rrThe electric near-field looks like that of an electric dipole

The magnetic near-field looks like that of a magnetic dipole


Wire Loop Antennas in Consumer Products

A 2 m loop antenna for 1-30 MHz operation

A 30 inch home made multiple turn loop antenna

A 10 cm loop antenna with a feed

10


Wire Loop Antennas in Medical Devices - II

A flexible “band-aid” chip for wireless EEG (Electroencephalography) measurements at 2.4 GHz with a loop antenna

A flexible “band-aid” chip for wireless EMG (Electromyography) measurements at 433 MHz with a loop antenna


Wire Loop Antennas for Everybody - III

A portable loop antenna for 5-10 MHz operation on somebody’s van

A home made loop antenna in somebody’s backyard

Mine is bigger – says this guy!

lecture 31 wire antennas - cornell university 31 wire antennas in this lecture you will learn: •...

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