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CRL 715 2009-10 CRL 715 2009-10 Radiating Systems for RF Radiating Systems for RF Communications Communications 1 • Aperture antennas constitute a large class of antennas, which emit EM waves through an opening (or aperture). • These antennas have close analogs in acoustics: the megaphone and the parabolic microphone . The pupil of the human eye is a typical aperture receiver for optical radiation. At radio and microwave frequencies, horns, waveguide apertures and reflectors are examples of aperture antennas. • Aperture antennas are commonly used at UHF and above as their gain increases as f 2 (approximately). For an efficient and highly directive aperture antenna, its area should be comparable or larger than λ 2 . Hence, these antennas are impractical at low frequencies. Aperture Antennas Aperture Antennas

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Page 1: Aperture 2K9 10

CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 11

• Aperture antennas constitute a large class of antennas, which emit EM waves through an opening (or aperture).

• These antennas have close analogs in acoustics: the megaphone and the parabolic microphone. The pupil of the human eye is a typical aperture receiver for optical radiation. At radio and microwave frequencies, horns, waveguide apertures and reflectors are examples of aperture antennas.

• Aperture antennas are commonly used at UHF and above as their gain increases as f2 (approximately). For an efficient and highly directive aperture antenna, its area should be comparable or larger than λ2. Hence, these antennas are impractical at low frequencies.

Aperture AntennasAperture Antennas

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 22

• Another positive feature of the aperture antennas is their near-real valued input impedance and geometry compatibility with waveguide feeds.

• The radiation characteristics of wire antennas can be determined from the current distribution on the wire.

• If the current distribution is not known, alternate methods are necessary to compute the radiation characteristics e.g. from the fields on or in the vicinity of the antenna.

• One such technique is the “Field Equivalence Principle”.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 33

Uniqueness theorem

A solution is said to be unique if it is the only one possible among a given class of solutions.

• The EM field in a given region V is uniquely defined if:

a. all sources are given;

b. either the tangential Eτ components or the tangential Hτ components are specified at

the boundary S.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 44

Equivalence principle

• The equivalence principle follows from the uniqueness theorem. It allows for building simpler problems.

• As long as the equivalent problem preserves the boundary conditions of the original problem for the field at S, it is going to produce only one possible solution for the region outside S.

• Through equivalence principle, the fields outside an imaginary closed surface are obtained by placing over the closed surface suitable electric and magnetic current densities which satisfy the boundary conditions.

• The current densities are chosen so that the fields inside the closed surface are zero and outside they are equal to the fields produced by the actual sources.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 55

E1,H1

V1

1,1 1,1

E1,H1

V2

S

Original problem

J1 M1

The primary task is to replace the original problem by an equivalent problem which yields the same fields E1 and H1 outside S (within V2).

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 66

E,H

V1

1,1

1,1

E1,H1

V2

S

Equivalent problem

1ˆsM n E E

1ˆsJ n H H

Ms

Js

• The original sources J1 and M1 are removed, and assume there exist fields E and H inside S and fields E1 and H1 outside of S.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 77

• For this, the fields must satisfy the boundary conditions on tangential electric and magnetic fields components. Hence on imaginary surface there must exist equivalent sources:

• These current densities are said to be equivalent only within V2 as they produce same fields outside.

• Since the fields E and H within S are not of interest, they can be assumed to be zero:

1

1

ˆ

ˆ

s

s

J n H H

M n E E

1 10

1 10

ˆ ˆ|

ˆ ˆ|

s H

s E

J n H H n H

M n E E n E

Love’s Equivalence Principle

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 88

We can assume that the boundary S is a perfect electric conductor. This eliminates the surface electric currents, i.e., Js=0, and leaves just surface magnetic currents Ms, which radiate in the presence of a perfect electric surface.

We can apply Love’s equivalence principle in three different ways

E,H=0

V1

1,1

E1,H1

V2

S n̂

1ˆ 0sJ n H

1ˆsM n E

Electric conductor

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 99

E,H=0

V1

1,1

E1,H1

V2

S n̂

1ˆsJ n H

1ˆ 0sM n E

Magnetic conductor

We can assume that the boundary S is a perfect magnetic conductor. This eliminates the surface magnetic currents, i.e. Ms=0, and leaves just surface electric currents Js, which radiate in the presence of a perfect magnetic surface.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 1010

Make no assumptions about the materials inside S, and define both Js and Ms currents, which are radiating in free space (no fictitious conductors behind them). It can be shown that these equivalent currents create zero fields inside S.

• All three approaches lead to the same field solution according to the uniqueness theorem.

• The first two approaches are not very accurate in the general case of curvilinear boundary surface S. However, in the case of flat infinite planes (walls), the image theory can be used to reduce the problem to an open one.

• Image theory can be successfully applied to curved surfaces provided the curvature’s radius is large compared to the wavelength.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 1111

,

1ˆsM n E

1ˆsM n E

,

sM

,

1ˆ2sM n E

, ,

Equivalent problem - Electric Wall

Equivalent problem - images

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 1212

• Example: A waveguide aperture is mounted on an infinite ground plane. Assuming that the tangential component of the electric field over the aperture is Ea. Find an equivalent problem that will yield the same fields E, H radiated by the aperture to the right side of the interface.

aE

S

,

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 1313

S

n̂, ,

0

s

s

J

M

ˆ

s

s a

J

M n E

0

s

s

J

M

S

n̂, ,

0

0

s

s

J

M

0

ˆ

s

s a

J

M n E

0

0

s

s

J

M

S

n̂, ,

0

0

s

s

J

M

0

ˆ

s

s a

J

M n E

0

0

s

s

J

M

ˆs aM n E

(Image)

S

n̂, ,

0

0

s

s

J

M

0

ˆ2

s

s a

J

M n E

0

0

s

s

J

M

Solution:

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 1414

Radiation Equations

'

'

R

r'r

x’,y’,z’ on S

'cosR r r

R r

ψ is the angle between r and r’

Primed coordinates: sources

Unprimed coordinates: observation point

'4

'4

jkR

V

jkR

V

eA J dv

R

eF M dv

R

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 1515

'cos

'cos

''4 4

'4 4

'

jkR jkjkr

s

S

r

s

S

jkjkr

s

S

R jkr

s

S

N N J e dse e

A J dsR r

e eF M ds

R rL L M e ds

The far-zone fields and vector potentials are related as:

ˆ ˆ

ˆ ˆ

farA

farF

E j A A

H j F F

1far far far farF F A AE H r H E r

Since

ˆ ˆfarE j A F A F

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 1616

2 2k

0

4

4

r r

jkr

jkr

E H

jkeE L N H

r

jkeE L N H

r

'cos

'cos

'cos

'cos

cos cos cos sin sin '

sin cos '

cos cos cos sin sin '

sin cos '

jkrx y z

S

jkrx y

S

jkrx y z

S

jkrx y

S

N J J J e ds

N J J e ds

L M M M e ds

L M M e ds

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 1717

r

R

'r

x

z

y

Rectangular apertures in different planes(Example: y-z plane)

Nonzero components of Js and Ms are:

Jy, Jz, My Mz

ˆ'cos '

ˆ ˆ ˆ ˆ ˆ' ' sin cos sin sin cos

'sin sin 'cos

r

y z x y z

r r a

a y a z a a a

y z

' ' 'ds dy dz

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 1818

Uniform distribution on an infinite ground plane

0ˆ 2 ' 2 2 ' 2a yE a E a x a b y b

Find the far fields radiated by a rectangular aperture on an infinite ground plane with following field over the opening:

Fields radiated by the aperture:

0

0

0

sin sinsin

2

sin sin

sin cos sin

cos c

sin

os

2

2

2

r r

jkr

jkr

E H

abkE e X YE j H

r X Y

abkE e X YE j H

r X Y

ka kbX Y

sin cos

2

sin sin2

xx e

e dx

kaX

kbY

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 1919

The total-field amplitude pattern is, therefore,

2 2 2 2 2sin sin sin sinsin cos cos 1 cos sin

X Y X YE

X Y X Y

The E-plane (yz-plane: =/2) pattern

0

sin sin2

2 sin2

jkr

kbabkE e

E jkbr

The H-plane (xz-plane: =0) pattern

0

sin sin2

cos2 sin

2

jkr

kaabkE e

E jkar

Function of dimension ‘b’ (along y-axis)

Function of dimension ‘a’ (along x-axis)

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 2020

3 2a b

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 2121

• For electrically large apertures, the main beam is narrow and the square-root term is negligible, i.e., it is roughly equal to 1 for all observation angles within the main beam.

• That is why, in the theory of large arrays, it is assumed that the amplitude pattern of a rectangular aperture is

2 2 2 2 2sin sin sin sinsin cos cos 1 cos sin

X Y X YE

X Y X Y

sin sinX Y

X Y

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 2222

H-plane pattern for a=20 E-plane pattern for b=10

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 2323

First Null beamwidth for E-plane pattern:

For nulls to occur:

1114.6sin deg 1,2,3,....n

nn

b

n=1 gives FNBW

1 1

sin |2

2sin 57.3sin deg

n

n

kbn

n n

kb b

Beamwidth between nulls:

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 2424

Half-power beamwidth (E-plane):

Half-power point:

1

sin sin12

sin | 1.39122sin

2

0.44357.3sin deg

h

h

kbkb

kb

b

Half-power beamwidth:

1 0.443114.6sin degh b

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 2525

Maximum of first sidelobe:

1 1.43sin | 4.494 57.3sin deg

2 s s

kb

b

Total beamwidth between first sidelobes:

1 1.43114.6sin degs b

Maximum at the first sidelobe:

sin 4.4940.217 13.26

4.494sE dB

• When evaluating side-lobe levels and beamwidths in the H-plane, one has to include the cosθ factor, too.

• The larger the aperture, the less important this factor is.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 2626

Directivity:

max0 2

0 2 2

4 4

4 4rad

p em

UD ab

P

D A A

The physical (Ap) and the effective (Aem) areas of a uniform aperture are equal.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 2727

The tapered rectangular aperture on a ground plane

• The uniform rectangular aperture has the maximum possible effective area (for an aperture-type antenna) equal to its physical area it has the highest possible directivity for all constant-phase excitations of a rectangular aperture.

• The uniform distribution excitation produces the highest SLL of all constant-phase excitations of a rectangular aperture.

• A reduction of the SLL can be achieved by tapering the equivalent source distribution from a maximum at the aperture’s center to zero values at its edges.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 2828

• One practical aperture of tapered source distribution is the open rectangular waveguide. The dominant TE10 mode has the following distribution:

0ˆ cos ' 2 ' 2 2 ' 2a yE a E x a x a b y ba

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 2929

02

2

02

2

sin cos sin sin2

0

cos sinsin

2 2

2

cos sincos cos

2

2

2

2

r r

jkr

jkr

E H

abkE e X YE j H

r YX

abkE e X YE j H

r

kbY

Y

kX

X

a

Fields radiated by the aperture:

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 3030

The E-plane (yz-plane: =/2) pattern

sin sin2

sin2

n

kb

Ekb

The H-plane (xz-plane: =0) pattern

2 2

sin sin2

cos

sin2 2

n

ka

Eka

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 3131

H-PLANE PATTERN – UNIFORM VS. TAPERED ILLUMINATION (a = 3λ):

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 3232

Aperture efficiency

Directivity:

0 2 2

4 40.81

0.81

p em

em p

D

A A

A A

In general,

0 1em ap p apA A

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 3333

Example: Consider an aperture in an infinitely conducting ground plane having a length a and width b. The aperture field has a uniform amplitude but a 180° phase difference in the two halves of the aperture as shown below:

Find the far field of the antenna applying the radiation equation of a rectangular aperture.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 3434

2 20'cos 'cos

0 0

2 2 0

0

0

0

0 'cos 'sin cos 'sin

cos cos 2 ' ' 2 ' '

sin s2

2 cos cos

ˆ ˆ2 0.5 0 0.5 0.5

0 0.5 0.5 0.5ˆ

sin

ˆ2

b ajkr jkr

b a

z y

s

z y

a E a a x b y bM

x a b y ba E a

N N r x y

L E e dx dy E e dx dy

kb

L E b

20'sin cos 'sin cos

2 0

in sin' '

sin sin2

ajkx jkx

a

e dx e dxkb

2

0

2

0

sin 2sin2 cos cos

2

sin 2sin2 sin

2

XYL j E ab

Y X

XYL j E ab

Y X

sin cos2

sin sin2

xx e

e dx

kaX

kbY

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 3535

4

4

jkr

jkr

jkeE L

r

jkeE L

r

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 3636

Example (Problem 12.22 Balanis): A rectangular aperture with the dimensions a=λ/2 and b=λ/4 and is mounted on an infinite ground plane. What are the corresponding directivities (in dB) when the aperture has:

i) a triangular electric field distribution and an aperture efficiency of 75%, ii) a cosine-square electric field distribution and an aperture efficiency of 66%?

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 3737

Example: A rectangular aperture, of dimensions a and b, is mounted on an infinite ground plane, as shown in the figure. Assuming the tangential field over the aperture given by:

0ˆ 2 ' 2 2 ' 2a zE a E a y a b z b

Find the far-zone spherical electric and magnetic field components radiated by the aperture.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 3838

Horn Antennas Horn antennas are popular in the microwave band (above 1 GHz). Horns provide high gain, low VSWR (with waveguide feeds),

relatively wide bandwidth, and they are not difficult to make. There are three basic types of rectangular horns.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 3939

• The horns can be also flared exponentially. This provides better matching in a broad frequency band, but is technologically more difficult and expensive.

• The rectangular horns are ideally suited for rectangular waveguide feeders.

• The horn acts as a gradual transition from a waveguide mode to a free-space mode of the EM wave.

• When the feeder is a cylindrical waveguide, the antenna is usually a conical horn.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 4040

• Why is it necessary to consider the horns separately instead of applying the theory of waveguide aperture antennas directly to the aperture of the horn?

• It is because the so-called phase error occurs due to the difference between the length from the center of the feeder to the center of the horn aperture and the length from the center of the feeder to the horn edge.

• This complicates the analysis, and makes the uniform-phase aperture results invalid for the horn apertures.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 4141

E-Plane sectoral Horn

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 4242

lE=e

R0=R=1

E=ψE

B=b1

21

21

21

' 2

1

' 2

1

' 21

1

' ' ' 0

'( ', ') cos '

'( ', ') sin '

'( ', ') cos '

cos

z x y

j ky

y

j ky

z

j ky

x

e e

E E H

E x y E x ea

H x y jE x eka a

EH x y x e

a

( ')1

( ')1

cos '

cos '

jk yy

jk yy

EJ x e

a

J E x ea

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 4343

21

21

21 11 22 2

21 11 22 2

cos2

sin 1 cos ,8

2 2

cos2

cos 1 cos ,8

2 2

y

y

xjkr

j k k

x

xjkr

j k k

x

k aa k E e

E j e F t tr k a

k aa k E e

E j e F t tr k a

1 2 2 1 2 1

1 11 1 2 1

1 1

sin cos sin sin

,

1 1

2 2

x y

y y

k k k k

F t t C t C t j S t S t

kb kbt k t k

k k

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 4444

E-Plane sectoral HornE-Plane sectoral Horn

E-Plane ( )E-Plane ( )2

2

12

sin 21 1 ' '1 2

' 11 1

1

' 12 1

1

0

21 cos ,

8

sin2

sin2

r

jkrj k

E E

a k E eE j e F t t

r

bkt

bkt

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 4545

H-Plane ( )H-Plane ( )0

1 1 " "1 22 2

" 11

1

" 12

1

0

cos sin2

1 cos ,8

sin2 2

2

2

r

jkr

E E

kaa k E e

E j F t tr ka

bkt

bkt

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 4646

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 4848

The magnitude of the normalized pattern, excluding the The magnitude of the normalized pattern, excluding the factor , can be written as:factor , can be written as: 1 cos

' ' ' ' ' '1 2 2 1 2 1

' 11 1

1

21 1 1

21 1

21 1

1

,

sin2

812 1 sin

8 4

1 12 1 sin

4 8

nE F t t C t C t j S t S t

bkt

b b

b

b bs s

s

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 4949

' 12 1

1

21 1 1

21 1

1

sin2

812 1 sin

8 4

1 12 1 sin

4

bkt

b b

b

bs

s

For a given value of , the normalized field can be plotted For a given value of , the normalized field can be plotted as a function of . as a function of .

These plots are usually referred to as These plots are usually referred to as universal curvesuniversal curves..

1 sinb s

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 5050

From the universal curves, the E-plane pattern of any E-From the universal curves, the E-plane pattern of any E-plane sectoral horn can be obtained.plane sectoral horn can be obtained.

First determine the value of . For that value of , the field First determine the value of . For that value of , the field strength (in dB) as a function of is obtained from strength (in dB) as a function of is obtained from the following figure. Finally, the value of , the following figure. Finally, the value of , normalized to 0 dB and written as , is normalized to 0 dB and written as , is added to that number to get the required field strength.added to that number to get the required field strength.

s s 1 sinb

1 cos 1020log 1 cos 2

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 5252

Directivity of E-plane sectoral Horn:Directivity of E-plane sectoral Horn:

2 2max 1 1 1

1 1 1

4 64

2 2E

rad

U a b bD C S

P b

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 5353

If the values of , which correspond to the If the values of , which correspond to the maximum directivities are plotted versus their maximum directivities are plotted versus their corresponding values of , it can be shown that corresponding values of , it can be shown that each optimum directivity occurs wheneach optimum directivity occurs when

The corresponding value of is:The corresponding value of is:

1b

1

1 12b b

s

1

4s

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 5454

H-Plane Sectoral HornH-Plane Sectoral Horn

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 5555

Aperture Fields:Aperture Fields:

' '

''2

1

'' 2

1

2

22

0

' cos '

' cos '

1 '' cos

2

x y

jk xy

jk xx

h h

E H

E x E x ea

EH x x e

a

xx

Note: The aperture fields for E-plane sectoral horn were functions of x’ and y’, whereas for H-plane sectoral horn, they are functions of x’ only

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Radiated FieldsRadiated Fields: Over the aperture the equivalent : Over the aperture the equivalent current densities arecurrent densities are

'2

1

'2

1

0

cos '

cos '

x z y z

jk xy

jk xx

J J M M

EJ x e

a

M E x ea

Note: The equivalent current densities for E-plane sectoral horn and H-plane sectoral horns are same except the complex exponential term and the constant.

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Radiated FieldsRadiated Fields

'coscos cos cos sin sin 'jkrx y z

S

N J J J e ds

From the radiation equations for aperture, we have

So, in the present case this becomes,

'coscos sin 'jkry

S

N J e ds

For aperture in the XY plane,

'cos 'sin cos 'sin sin

' ' '

r x y

ds dx dy

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 5858

Hence,

21 2

2'sin sin

12

2' 'sin cos

22

cos sin

sin sin sin2

'sin sin

2

cos ' '

bjky

b

ajk x x

a

EN I I

kb

I e dy bkb

I x e dxa

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 5959

1 222 1 2 1 2

1 2 2 1 2 1

2 22 2

1 2

1 11 2 2 2

2 2

11 2

2

cos sin sin', ' ", "

2

,

' "sin sin

2 2 2

1 1' ' ' '

2 2

1" "

2

jf jf

x x

x x

x

b YN E e F t t e F t t

k Y

F t t C t C t j S t S t

k k kbf f Y

k k

ka kat k t k

k k

kat k

k

12 2

2

1" "

2

' sin cos " sin cos

x

x x

kat k

k

k k k ka a

Finally,

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 6060

1 2

1 2

1 2

22 1 2 1 2

22 1 2 1 2

22 1 2 1 2

22

cos sin sin', ' ", "

2

cos sin', ' ", "

2

cos cos sin', ' ", "

2

sin

2

jf jf

jf jf

jf jf

b YN E e F t t e F t t

k Y

b YN E e F t t e F t t

k Y

b YL E e F t t e F t t

k Y

bL E

k

1 21 2 1 2

sin', ' ", "jf jfY

e F t t e F t tY

Similarly can be found., ,N L L

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 6161

The far-zone electric field components can then be The far-zone electric field components can then be found out:found out:

04 4

jkr jkr

rke ke

E E j L N E j L Nr r

1 2

1 2

22

1 2 1 2

22

1 2 1 2

0

8

sinsin 1 cos ', ' ", "

8

sincos 1 cos ', ' ", "

r

jkr

jf jf

jkr

jf jf

E

kb eE jE

r

Ye F t t e F t t

Y

kb eE jE

r

Ye F t t e F t t

Y

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The Electric Field in the E-Plane (y-z plane)2

1 2

22

1 2 1 2

1

1

0

8

sin1 cos ', ' ", "

sin2

'

"

r

jkr

jf jf

x

x

E E

kb eE jE

r

Ye F t t e F t t

Y

kbY

ka

ka

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The Electric Field in the H-Plane (x-z plane)0

1 2

22

1 2 1 2

1

1

0

8

1 cos ', ' ", "

' sin

" sin

r

jkr

jf jf

x

x

E E

kb eE jE

r

e F t t e F t t

k ka

k ka

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The Directivity of a H-plane sectoral HornThe Directivity of a H-plane sectoral Horn

2 22

1

2 1

1 2

2 1

1 2

4

1

2

1

2

Hb

D C u C v S u S va

au

a

av

a

The Directivity is optimum whenThe Directivity is optimum when

1 23a

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Pyramidal HornPyramidal Horn

Most widely used horn.Most widely used horn.

Flared in both directions.Flared in both directions.

Radiation characteristics Radiation characteristics are combination of the E- are combination of the E- and H-plane sectoral and H-plane sectoral horns.horns.

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 6767

The tangential components of the E- and H-fields over the The tangential components of the E- and H-fields over the aperture of the horn are approximated by:aperture of the horn are approximated by:

2 22 1

2 22 1

' ' 2

01

' ' 20

1

' ', ' cos '

' ', ' cos '

j k x y

y

j k x y

x

E x y E x ea

EH x y x e

a

The equivalent current densities (approximated):The equivalent current densities (approximated):

2 22 1

2 22 1

' ' 20

1

' ' 2

01

', ' cos '

', ' cos '

j k x y

y

j k x y

x

EJ x y x e

a

M x y E x ea

We have cosinusoidal amplitude distribution in the x’ We have cosinusoidal amplitude distribution in the x’

direction and quadratic phase variations in both x’ and y’ direction and quadratic phase variations in both x’ and y’ directions.directions.

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One can derive the following expressions from the current One can derive the following expressions from the current densities:densities:

0 01 2 1 2

0 1 2 0 1 2

cos sin cos

cos cos sin

E EN I I N I I

L E I I L E I I

22

22

22

' 2 'sin cos21 2

' 222 1 2 1

" 22 1 2 1

cos ' '

1' ' ' '

2

" " " "

x

x

jk x xa

a

j k k

j k k

I x e dxa

e C t C t j S t S tk

e C t C t j S t S t

21

21

' 2 'sin sin22 2

212 1 2 1

'

y

jk y yb

b

j k k

I e dy

e C t C t j S t S tk

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CRL 715 2009-10CRL 715 2009-10 Radiating Systems for RF CommunicationsRadiating Systems for RF Communications 6969

Far-zone E-field components are:Far-zone E-field components are:

01 2

01 2

0

sin 1 cos4 4

cos 1 cos4 4

r

jkrjkr

jkrjkr

E

kE ekeE j L N j I I

r r

kE ekeE j L N j I I

r r

The principal E-plane pattern of a pyramidal horn, aside The principal E-plane pattern of a pyramidal horn, aside from a normalization factor, is identical to the E-plane pattern from a normalization factor, is identical to the E-plane pattern of an E-plane sectoral horn.of an E-plane sectoral horn.

The principal H-plane pattern of a pyramidal horn, aside The principal H-plane pattern of a pyramidal horn, aside from a normalization factor, is identical to the H-plane pattern from a normalization factor, is identical to the H-plane pattern of an H-plane sectoral horn.of an H-plane sectoral horn.

Hence, the pattern is very narrow in both the principal planes.Hence, the pattern is very narrow in both the principal planes.

2

0

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3-D Field Pattern of a Pyramidal Horn3-D Field Pattern of a Pyramidal Horn

The maximum radiation is not necessarily directed along its axis, because the phase error taper at the aperture is such that the rays coming from different parts of the aperture towards the axis are not in phase and do not add constructively.

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To physically construct a pyramidal horn, the To physically construct a pyramidal horn, the dimension and should be equal:dimension and should be equal:ep hp

1 22

11

1 22

11

1

4

1

4

ee

hh

p b bb

p a aa

The method outlined above gives accurate patterns for The method outlined above gives accurate patterns for angular regions near the main lobe and its closest minor angular regions near the main lobe and its closest minor lobes. The accurately predict the field intensity in the minor lobes. The accurately predict the field intensity in the minor lobes, diffraction techniques (accounting for diffraction near lobes, diffraction techniques (accounting for diffraction near the aperture edges) are utilized. Diffraction becomes the aperture edges) are utilized. Diffraction becomes dominant in regions where the radiation is of very low dominant in regions where the radiation is of very low intensity.intensity.

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Co- and Cross-PolarizationCo- and Cross-Polarization

The patterns that are plotted represent the main The patterns that are plotted represent the main polarization of the field radiated by the antenna. This is polarization of the field radiated by the antenna. This is called co-polarized pattern.called co-polarized pattern.

Ideally the radiated field has no orthogonal (to the main Ideally the radiated field has no orthogonal (to the main polarization) component (cross-polarized), provided the polarization) component (cross-polarized), provided the antenna is symmetrical and is excited in the dominant antenna is symmetrical and is excited in the dominant mode.mode.

In practice, because of nonsymmetries, defects in In practice, because of nonsymmetries, defects in construction or excitation of higher order modes, all construction or excitation of higher order modes, all antennas have cross-polarized components (usually of very antennas have cross-polarized components (usually of very low intensity).low intensity).

For good designs, cross-polarized components should be at For good designs, cross-polarized components should be at least 30 dB below the co-polarized.least 30 dB below the co-polarized.

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Directivity of a Pyramidal HornDirectivity of a Pyramidal Horn

2

2 21 1 1

1 1 1

2 22

1

2 1

1 2

2 1

1 2

32

64

2 2

4

1

2

1

2

p E H

E

H

D D Dab

a b bD C S

b

bD C u C v S u S v

a

au

a

av

a