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School of Electrical Engineering

Lecture 4: Aperture antennas

Aperture antennas

KTH School of Electrical Engineering • www.ee.kth.se 2

• They are antennas in which we know the distribution of the fields in asurface (typically flat).

• They are typically employed for high power applications, and spaceapplications.

Guided waves

Radiation

Aperture

Horn antennas


• To increase the directivity of the radiated antenna,normally, a horn is employed.

Horn antennas


• Electric field in a horn antenna:

- Transition between waveguide and horn.

Side view Top view

TE10 mode

Equivalence theory


Our Problem Equivalent problem

Equivalence principle models


Love’s equivalent Electric conductor equivalent

Magnetic conductor

equivalent

][ˆ

][ˆ

1

1

EEnM

HHnJ

S

S

Theory


• The radiation can be derived from:

1. Magnetic potential vector:

2. Electric potential vector:

'

' '4

S

jkR

S dSR

eJA

'4

'

'

S

jkR

S dSR

eMF

Theory


• In far-field: cos'rrR Phase

rR Amplitude

Nr

edSeJ

r

edS

R

eJA

jkr

S

jkr

S

jkr

S

jkR

S

4'

4'

4'

cos'

'

Lr

edSeM

r

edS

R

eMF

jkr

S

jkr

S

jkr

S

jkR

S

4'

4'

4'

cos'

'

''

cos'

S

jkr

S dSeML

'

cos' 'S

jkr

S dSeJN

Theory


• Again, since we are in the far field, we will have onlycomponents θ and ϕ:

FjE

FjE

AjE

AjE

F

F

A

A

FjH

FjH

AjH

AjH

F

F

A

A

FHj

FAjAjEEE AFA

1111

FFA Ej

AFjFjAHHH

1111

Fields


• The fields will be:

jkr

jkr

r

eL

Nr

kjH

eL

Nr

kjH

H

4

4

0

jkr

jkr

r

eNLr

kjE

eNLr

kjE

E

4

4

0Nr

eA

jkr

4

Lr

eF

jkr

4

FjE

FjE

AjE

AjE

F

F

A

A

FjH

FjH

AjH

AjH

F

F

A

A

Fields


• Where our N and L components will be obtained as:

'cossin

'sinsincoscoscos

'

cos'

'

cos'

S

jkr

yx

S

jkr

zyx

dSeMML

dSeMMML

'cossin

'sinsincoscoscos

'

cos'

'

cos'

S

jkr

yx

S

jkr

zyx

dSeJJN

dSeJJJN

Summary


1. To select a closed (imaginary) surface withknown and

2. To obtain L and N.

3. To obtain the fields: E and H.

4. To determine the radiated fields (far fields)

SM

SJ

Uniform distribution


• Lets assume a uniform distribution in the aperture over a ground plane:

yEEaˆ

0

2'

2

2'

2

by

b

ax

a

Elsewhere

by

b

ax

a

ExEyzEnM aperture

S

02

'2

2'

22ˆˆˆ2ˆ2 00

0SJ

1) M and S



Elsewhere

by

b

ax

a

ExEyzEnM aperture

S

02

'2

2'

22ˆˆˆ2ˆ2 00

0SJ

2) L and N0 NN

2/

2/

2/

2/

cos'

0

2/

2/

2/

2/

cos'

0

''sin2

''coscos2

b

b

a

a

jkr

b

b

a

a

jkr

dydxeEL

dydxeEL

'cossin

'sinsincoscoscos

'

cos'

'

cos'

S

jkr

yx

S

jkr

zyx

dSeMML

dSeMMML



2) L and N

2/

2/

2/

2/

cos'

0

2/

2/

2/

2/

cos'

0

''sin2

''coscos2

b

b

a

a

jkr

b

b

a

a

jkr

dydxeEL

dydxeEL

2/

2/

2/

2/

)sinsin'cossin'(

0

2/

2/

2/

2/

)sinsin'cossin'(

0

''sin2

''coscos2

b

b

a

a

yxjk

b

b

a

a

yxjk

dydxeEL

dydxeEL

2

2sin

'

2/

2/

de j

sinsinsin2

sinsincoscos2

0

0

abEL

abEL

sinsin2

cossin2

kb

ka

sinsin'cossin'

)ˆcosˆsinsinˆcos(sin)ˆ'ˆ'(ˆ'cos'

yx

zyxyyxxrrr



3) E and H

jkrjkr

jkrjkr

r

er

abkEje

L

r

kjH

er

abkEje

L

r

kjH

H

sinsin

2

sin

4

sinsin

2

coscos

4

0

0

0

jkrjkr

jkrjkr

r

er

abkEjeL

r

kjE

er

abkEjeL

r

kjE

E

sinsin

2

coscos

4

sinsin

2

sin

4

0

0

0

0 NN



4) Far field (E-plane)

jkr

r

ekb

kb

Er

abkjH

H

H

sin2

sin2

sin

2

0

0

0

0

sin2

sin2

sin

2

0

0

E

ekb

kb

Er

abkjE

E

jkr

r

2/

4) Far field (H-plane) 0

jkr

r

eka

ka

Er

abkjE

E

E

sin2

sin2

sin

cos2

0

0

0

0

sin2

sin2

sin

cos2

0

0

H

eka

ka

Er

abkjH

H

jkr

r

TEM Mode:

transversal electric

and magnetic

H

EZwave

H

EZwave

H

ErZwave

ˆ

Uniform distribution: Radiation pattern


abAD aperture 22max

44

Other distributions


Other shapes: Circular


Horns


• The are normally used to increase the directivity of an aperture.

• The are fed with a waveguide.

• Normally employed as feeding of reflectors.

waveguide horn

A

L

tan α = A/(2L)α

Horns


• Lets assume a TE10 mode in the waveguide.

• The amplitude in the horn will be the same, but wewill have a phase error between the center and thecorners.

ljk

waveguidehorn eEE

Error in the phase-Constant in E-plane

-Cosine in H-plane

yxa

EEwaveguideˆcos1

yx

Horns: Phase error


waveguide horn

A

Ltan α = A/(2L)

α

∆l

222][ yLlL

The phase error:

L

Al

8

2

max

Normally,

given in rad: ][max radlk

The maximum error:

L

yLyLl

2

222

y

yexa

EE ljk

hornˆcos1

Horns: Types


E-planeH-plane

Pyramidal Conical

yexa

EEL

yjk

hornˆcos

2

1

2

yexa

EEL

xjk

hornˆcos

2

1

2

yeexa

EEL

yjk

L

xjk

hornˆcos

22

1

22

Corrugated horns


• The waveguide presentscorrugations.

• These corrugations areemployed to obtain a moresymmetric radiation pattern,to reduce the diffraction inthe borders and x-polarization.

Homework


• We have a X-band waveguide, and we are operatingat 10GHz.

• We would like to design a E-plane sectoral horn inwhich our maximum phase error is 50º. The increaseof the size will be to the double in E-plane.

• Calculate the required flare angle α of our horn.

Documents

School of Eletrical Engineering - KTH Antennas.pdf · KTH School of Electrical Engineering • 2 •They are antennas in which we know the distribution of the fields in a ... jkr