L18 Horns
L18 Horns
L18 Horns
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
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is usually a conical horn.
Why is it necessary to consider the horns separately instead of applying the
theory of waveguide aperture antennas directly? It is because the so-called
phase error occurs due to the difference between the lengths from the center of
the feeder to the center of the horn aperture and the horn edge. This makes the
uniform-phase aperture results invalid for the horn apertures.
lH
R x
R0
a A
H z
RH
2
A
lH2 R02 , (18.1)
2
A
H arctan , (18.2)
0
2 R
l 1
RH A a H . (18.3)
A 4
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The two fundamental dimensions for the construction of the horn are A and
RH .
The tangential field arriving at the input of the horn is composed of the
transverse field components of the waveguide dominant mode TE10:
E y E0 cos x e j g z
a (18.4)
H x Ey / Z g
where
Zg is the wave impedance of the TE10 mode;
2
1
2a
2
g 0 1 is the propagation constant of the TE10 mode.
2a
Here, 0 2 / , and is the free-space wavelength. The field that
is illuminating the aperture of the horn is essentially an expanded version of the
waveguide field. Note that the wave impedance of the flared waveguide (the
horn) gradually approaches the intrinsic impedance of open space , as A (the
H-plane width) increases.
The complication in the analysis arises from the fact that the waves arriving
at the horn aperture are not in phase due to the different path lengths from the
horn apex. The aperture phase variation is given by
e j ( R R0 ) . (18.5)
Since the aperture is not flared in the y-direction, the phase is uniform in this
direction. We first approximate the path of the wave in the horn:
x
2
1 x 2
R R02 x 2 R0 1 R0 1 . (18.6)
R0 2 R0
The last approximation holds if x R0 , or A / 2 R0 . Then, we can assume
that
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1 x2
R R0 . (18.7)
2 R0
Using (18.7), the field at the aperture is approximated as
j x2
Ea y E0 cos x e 2 R0 . (18.8)
A
The field at the aperture plane outside the aperture is assumed equal to zero.
The field expression (18.8) is substituted in the integral I yE (see Lecture 17):
I yE Ea y ( x, y)e j ( x sin cos y sin sin ) dxdy , (18.9)
SA
A /2 b /2
j x 2
I yE E0 cos x e 2 R0 e j x sin cos dx e j y sin sin dy . (18.10)
A b /2
A /2
I ( , )
The second integral has been already encountered. The first integral is
cumbersome and the final result only is given below:
b
sin sin sin
1 R0 2 ,
I y E0
E I ( , ) b (18.11)
2 b
sin sin
2
where
2
R0
sin cos
C ( s2 ) jS ( s2 ) C ( s1 ) jS ( s1 )
j
I ( , ) e 2 A
2
(18.12)
R
j 0 sin cos
e 2 A C (t2 ) jS (t2 ) C (t1 ) jS (t1 )
and
1 A R0
s1 R0 u ;
R0 2 A
1 A R
s2 R0 u 0 ;
R0 2 A
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1 A R0
t1 R0 u ;
R0 2 A
1 A R0
t2 R0 u ;
R0 2 A
u sin cos .
Ea y E0 cos x e E0e j R
j R02 x 2 R0
0 cos x e j R02 x 2
. (18.14)
A A
The far field can be now calculated as (see Lecture 17):
e j r
E j (1 cos )sin I yE ,
4 r
(18.15)
e j r
E j (1 cos )cos I yE ,
4 r
or
b
sin sin sin
R0 e j r 1 cos 2
E j E0b b
4 r 2 sin sin (18.16)
2
I ( , ) θˆ sin φˆ cos .
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The amplitude pattern of the H-plane sectoral horn is obtained as
b
sin sin sin
1 cos 2 I ( , ) .
E b (18.17)
2 sin sin
2
Principal-plane patterns
b
sin sin sin
1 cos 2
E-plane ( 90 ): FE ( ) b (18.18)
2 sin sin
2
It can be shown that the second factor in (18.18) is exactly the pattern of a
uniform line source of length b along the y-axis.
H-plane ( 0 ):
1 cos
FH ( ) f H ( )
2
(18.19)
1 cos I ( , 0)
2 I ( 0, 0)
The H-plane pattern in terms of the I ( , ) integral is an approximation, which
is a consequence of the phase approximation made in (18.7). Accurate value for
f H ( ) is found by integrating numerically the field as given in (18.14), i.e.,
A /2
x j
A e
R02 x2
f H ( ) cos e j x sin dx . (18.20)
A /2
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E- AND H-PLANE PATTERN OF H-PLANE SECTORAL HORN
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The directivity of the H-plane sectoral horn is calculated by the general
directivity expression for apertures (for derivation, see Lecture 17):
2
S Ea ds
4
D0 2 . A
(18.21)
| Ea |2 ds
S A
1 1
p1 2 t 1 , p2 2 t 1 ;
8t 8t
2
1 A 1
t .
8 R0 /
The factor t explicitly shows the aperture efficiency associated with the
aperture cosine taper. The factor phH is the aperture efficiency associated with
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R0 100
Stutzman
It can be shown that the optimal directivity is obtained if the relation between A
and R0 is
A 3 R0 , (18.24)
or
A R0
3 . (18.25)
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1.2 The E-plane sectoral horn
lE
R y
R0
b B
E z
RE
E-plane (y-z) cut of an E-plane
sectoral horn
The geometry of the E-plane sectoral horn in the E-plane (y-z plane) is
analogous to that of the H-plane sectoral horn in the H-plane. The analysis is
following the same lines as in the previous section. The field at the aperture is
approximated by [compare with (18.8)]
j y2
Ea y E0 cos x e 2 R0 . (18.26)
a
Here, the approximations
y
2
1 y 2
R R02 y 2 R0 1 R0 1 (18.27)
R0 2 R0
and
1 y2
R R0 (18.28)
2 R0
are made, which are analogous to (18.6) and (18.7).
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The radiation field is obtained as
2
R B
4a R0 e j r
j 0 sin sin
E j E0 e 2 2 θˆ sin φˆ cos
4 r
a
cos sin cos (18.29)
(1 cos ) C (r ) jS (r ) C (r ) jS (r ) .
2
2 2 1 1
2 a
2
1 sin cos
2
The arguments of the Fresnel integrals used in (18.29) are
B B
r1 R0 sin sin ,
R0 2 2
(18.30)
B B
r2 R0 sin sin .
R0 2 2
Principal-plane patterns
The normalized H-plane pattern is found by substituting 0 in (18.29):
a
cos sin
1 cos 2 .
H ( ) (18.31)
a
2
2
1 sin
2
The second factor in this expression is the pattern of a uniform-phase cosine-
amplitude tapered line source. (Prove!)
The normalized E-plane pattern is found by substituting 90 in
(18.29):
E ( )
1 cos
f E ( )
1 cos C (r2 ) C (r1 )2 S (r2 ) S (r1 )2 . (18.32)
2 2 4 C 2 (r 0 ) S 2 (r 0 )
Here, the arguments of the Fresnel integrals are calculated for 90 :
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B B
r1 R0 sin ,
R0 2 2
(18.33)
B B
r2 R0 sin ,
R0 2 2
and
B
r 0 r2 ( 0) . (18.34)
2 R0
Similar to the H-plane sectoral horn, the principal E-plane pattern can be
accurately calculated if no approximation of the phase distribution is made.
Then, the function f E ( ) has to be calculated by numerical integration of
(compare with (18.20))
B /2
R02 y 2
f E ( ) e j e j sin y dy . (18.35)
B /2
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E- AND H-PLANE PATTERN OF E-PLANE SECTORAL HORN
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Directivity
The directivity of the E-plane sectoral horn is found in a manner analogous
to the H-plane sectoral horn:
a 32 B 4
DE ph
E t ph
E aB , (18.36)
2
where
8 C 2 (q) S 2 (q) B
t , E , q .
2 ph
q2 2 R0
A family of universal directivity curves DE / a vs. B / with R0 being a
parameter is given below.
R0 100
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The optimal relation between the flared height B and the horn length R0 is
B 2 R0 . (18.37)
where
8
t ;
2
ph
H
2
64t
C ( p ) C ( p ) S ( p ) S ( p ) ;
1 2
2
1 2
2
2
1 1 1 A 1
p1 2 t 1 , p2 2 t 1 , t H ;
8t 8t 8 R0 /
C 2 (q) S 2 (q) B
ph
E , q .
q2 2 R0
E
The gain of a horn is usually very close to its directivity because the radiation
efficiency is very good (low losses). The directivity as calculated with (18.39)
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is very close to measurements. The above expression is a physical optics
approximation, and it does not take into account only multiple diffractions, and
the diffraction at the edges of the horn arising from reflections from the horn
interior. These phenomena, which are unaccounted for, lead to minor
fluctuations of the measured results about the prediction of (18.39). That is why
horns are often used as gain standards in antenna measurements.
The optimal directivity of an E-plane horn is achieved at q 1 [see also
(18.37)], ph
E 0.8 . The optimal directivity of an H-plane horn is achieved at
aperture efficiency of
ph
P H E 0.632 .
ph ph (18.40)
The total aperture efficiency includes the taper factor, too:
ph
P H E 0.81 0.632 0.51 .
t ph ph (18.41)
Therefore, the best achievable directivity for a rectangular waveguide horn is
about half that of a uniform rectangular aperture.
We reiterate that best accuracy is achieved if phH and E are calculated
ph
numerically without using the second-order phase approximations in (18.7) and
(18.28).
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y x
R0E R0H
b B a
E A z
H
RE
RH
Similarly, the maximum-gain condition for the H-plane of (18.24) together with
(18.43) yields
A a A2 ( A a)
RH A . (18.47)
A 3 3
Since RE RH must be fulfilled, (18.47) is substituted in (18.46), which gives
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1 8 A( A a )
B b b2 . (18.48)
2 3
Substituting in the expression for the horn’s gain
4
G ap AB , (18.49)
2
gives the relation between A, the gain G, and the aperture efficiency ap :
4 1 8 A(a a )
G ap A b b 2
, (18.50)
2 2 3
3bG 2 3G 2 4
A4 aA3 A 0. (18.51)
8 ap 32 2 ap
2
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The gain increases with frequency, which is typical for aperture antennas.
However, the curve shows saturation at higher frequencies. This is due to the
decrease of the aperture efficiency, which is a result of an increased phase
difference in the field distribution at the aperture.
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The pattern of a “large” pyramidal horn ( f 10.525 GHz, feeder is waveguide
WR90):
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Comparison of the E-plane patterns of a waveguide open end, “small”
pyramidal horn and “large” pyramidal horn:
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Note the multiple side lobes and the significant back lobe. They are due to
diffraction at the horn edges, which are perpendicular to the E field. To reduce
edge diffraction, enhancements are proposed for horn antennas such as
corrugated horns
aperture-matched horns
Corrugated horns taper the E field in the vertical direction, thus, reducing side-
lobes and diffraction from edges. The overall main beam becomes smooth and
nearly rotationally symmetrical (esp. for A B ). This is important when the
horn is used as a feed to a reflector antenna.
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Comparison of the H-plane patterns of a waveguide open end, “small”
pyramidal horn and “large” pyramidal horn:
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2 Circular apertures
2.1 A uniform circular aperture
The uniform circular aperture is approximated by a circular opening in a
ground plane illuminated by a uniform plane wave normally incident from
behind.
z
a
E y
Here, J 0 is the Bessel function of the first kind of order zero. Applying the
identity
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xJ 0 ( x)dx xJ1 ( x) (18.57)
to (18.56) leads to
a
I xE 2 E0 J1 ( a sin ) . (18.58)
sin
In this case, the equivalent magnetic current formulation of the equivalence
principle is used [see Lecture 17]. The far field is obtained as
e j r E
ˆ
E θ cos φˆ cos sin j
2 r
Ix
(18.59)
e j r 2 J1 ( a sin )
ˆ
θ cos φˆ cos sin j E0 a 2
2 r a sin
.
Principal-plane patterns
2 J1 ( a sin )
E-plane ( 0 ): E ( ) (18.60)
a sin
2 J1 ( a sin )
H-plane ( 90): E ( ) cos (18.61)
a sin
The larger the aperture, the less significant the cos factor is in (18.61)
because the main beam in the 0 direction is very narrow and in this small
solid angle cos 1. Thus, the 3-D pattern of a large circular aperture features
a fairly symmetrical beam.
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Example plot of the principal-plane patterns for a 3 :
1
E-plane
0.9 H-plane
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-15 -10 -5 0 5 10 15
6*pi*sin(theta)
The half-power angle for the f ( ) factor is obtained at a sin 1.6 . So,
the HPBW for large apertures ( a ) is given by
1.6 1.6
HPBW 21/2 2arcsin 2 58.4 , deg. (18.63)
a a 2a
For example, if the diameter of the aperture is 2a 10 , then HPBW 5.84.
The side-lobe level of any uniform circular aperture is 0.1332 (-17.5 dB).
Any uniform aperture has unity taper aperture efficiency, and its directivity
can be found directly in terms of its physical area,
4 4
Du Ap a2 . (18.64)
2 2
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2.2 Tapered circular apertures
Many practical circular aperture antennas can be approximated as radially
symmetric apertures with field amplitude distribution, which is tapered from
the center toward the aperture edge. Then, the radiation integral (18.56) has a
more general form:
a
I xE 2 E0 ( ) J 0 ( sin )d . (18.65)
0
In (18.65), we still assume that the field has axial symmetry, i.e., it does not
depend on . Often used approximation is the parabolic taper of order n:
n
2
Ea ( ) E0 1 (18.66)
a
where E0 is a constant. This is substituted in (18.65) to calculate the respective
component of the radiation integral:
n
a
2
I xE ( ) 2 E0 1 J 0 ( sin )d . (18.67)
0 a
The following relation is used to solve (18.67):
1
2n n!
(1 x 2 ) n xJ 0 (bx ) dx n 1 J n 1 (b) .
b
(18.68)
0
is the normalized pattern (neglecting the angular factors such as cos and
cos sin ).
The aperture taper efficiency is calculated to be
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2
1 C
C
n 1
t . (18.71)
2 C (1 C ) (1 C ) 2
C2
n 1 2n 1
Here, C denotes the pedestal height. The pedestal height is the edge field
illumination relative to the illumination at the center.
The properties of several common tapers are given in the tables below. The
parabolic taper ( n 1 ) provides lower side lobes in comparison with the
uniform distribution ( n 0 ) but it has a broader main beam. There is always a
trade-off between low side-lobe levels and high directivity (small HPBW).
More or less optimal solution is provided by the parabolic-on-pedestal aperture
distribution. Moreover, this distribution approximates very closely the real case
of circular reflector antennas, where the feed antenna pattern is intercepted by
the reflector only out to the reflector rim.
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