A design procedure for slotted waveguide antennas with specified sidelobe levels

Abstract

Slotted waveguide antenna (SWA) arrays offer clear advantages in terms of their design, weight, volume, power handling, directivity, and efficiency. For broad-wall SWAs, the slot displacements from the wall centerline determine the antenna's sidelobe level (SLL). This paper presents an inventive procedure for the design of broadwall SWAs with desired SLLs. For a specified number of identical longitudinal slots, this procedure finds the slots length, width, locations along the length of the waveguide, and displacements from the centerline. Illustrative examples, based on Taylor, Chebyshev and the binomial distributions are given. In these examples, elliptical slots are considered, since their rounded corners are more robust for high power applications.

Full text

A Design Procedure for Slotted Waveguide
Antennas with Specified Sidelobe Levels
H. M. El Misilmani1,2, M. Al-Husseini2, K. Y. Kabalan1, and A. El-Hajj1
1ECE Department, American University of Beirut, Beirut, Lebanon
2Beirut Research and Innovation Center, Lebanese Center for Studies and Research, Beirut, Lebanon
hilal.elmisilmani@ieee.org, husseini@ieee.org, kabalan@aub.edu.lb, elhajj@aub.edu.lb
Abstract—Slotted waveguide antenna (SWA) arrays offer clear
advantages in terms of their design, weight, volume, power han-
dling, directivity, and efficiency. For broad-wall SWAs, the slot
displacements from the wall centerline determine the antenna’s
sidelobe level (SLL). This paper presents an inventive procedure
for the design of broadwall SWAs with desired SLLs. For a
specified number of identical longitudinal slots, this procedure
finds the slots length, width, locations along the length of the
waveguide, and displacements from the centerline. Illustrative
examples, based on Taylor, Chebyshev and the binomial distribu-
tions are given. In these examples, elliptical slots are considered,
since their rounded corners are more robust for high power
applications.
Keywords—Slotted waveguide antenna arrays, high power
microwaves, binomial, Chebychev, Taylor (one-parameter)
I. INTRODUCTION
Slotted Waveguide Antennas (SWAs) [1] radiate energy
through slots cut in a broad or narrow wall of a rectangular
waveguide. They are attractive due to their design simplicity,
since the radiating elements are an integral part of the feed
system, that is the waveguide itself. This design simplicity
removes the need for baluns or matching networks. SWAs
also offer significant advantages in terms of weight, volume,
high power handling, high efficiency, and good reflection
coefficient [2]. Thus, they have been ideal solutions for many
radar, communications, navigation, and high power microwave
applications [3].
SWAs can be realized as resonant or non-resonant according
to the wave propagation inside the waveguide (respectively
standing or traveling wave) [4], [5]. The design of a resonant
SWA is generally based on the procedure described by Elliot
[4], [6], [7], by which the waveguide end is short-circuited at
a distance of a quarter-guide wavelength from the center of
the last slot, and the inter-slot distance is one-half the guide
wavelength. For rectangular slots, the slot length should be
about half the free-space wavelength. Slot shapes that avoid
sharp corners are more suitable for high power applications,
since sharp corners aggravate the electrical breakdown prob-
lems. Elliptical slots are then an excellent candidate for such
applications [8], [9].
The resulting sidelobe level (SLL) for antenna arrays is
related to the excitations of the individual elements. In SWAs,
the excitation of each slot is proportional to its conductance.
For the case of longitudinal slots in the broadwall of a waveg-
uide, a slot conductance is controlled by its displacement from
the broadface centerline [10]. Thus, for a desired SLL, the
corresponding set of slots displacements should be determined.
This paper outlines the procedural steps for designing
resonant SWAs with longitudinal slots cut in the broadwall.
There are general guidelines for determining the width of the
slots, their inter-spacing along the length of the waveguide, and
their distances from both the feed port and the shorted end.
For rectangular slots, the slot length is about half the free-
space wavelength. However, for elliptical slots, as the ones
used in this work, the exact length is to be optimized. The
main focus of the presented procedure is to determine the slot
displacements that result in a specified SLL, which is done
as follows: for a desired SLL, the conductances of the slots
are obtained from a certain distribution, Chebyshev, Taylor, or
Binomial; then an equation that relates these conductances to
the displacements from the centerline is used to deduce these
displacements. Optimized parameters of the used distribution
should be found to lead to the desired SLL. Several examples
are given in this paper to illustrate the presented procedure. An
S-band SWA with 10 elliptical slots is used for these examples.
For each example, results for the obtained displacements,
reflection coefficients and radiation patterns are presented.
II. CO NFI GU RATI ON A ND GE NE RA L GUIDELINES
The S-band WR-284 waveguide used in our illustrative
examples has dimensions a= 2.8400 and b= 1.3700. The
design is done for the 3 GHz frequency. Ten elliptical slots
are made to one broadwall. The waveguide is shorted at one
end and fed at the other.
A. Slots Longitudinal Positions
The are general rules for the longitudinal positions of the
slots on the broadwall:
•The center of the first slot, Slot1, is placed at a distance
of quarter guide wavelength, or λg/4, from the the
waveguide feed,
•The center of the last slot, Slot10, is placed at λg/4from
the waveguide short-circuited side,
•The distance between the centers of two consecutive slots
is λg/2.

A guide wavelength is defined as the distance between
two equal phase planes along the waveguide. It is a function
of the operating wavelength (or frequency) and the lower
cutoff wavelength, and is calculated according to the following
equation:
λg=λ0
r1−λ0
λcutoff 
=c
f×1
r1−c
2a.f 
(1)
where λ0is the free-space wavelength calculated at 3 GHz,
and cis the speed of light. In this case, λg= 138.5mm.
Based on the above guidelines, the total length of the SWA
is 5λg, as shown in Fig. 1.
Figure 1. Slotted waveguide with 10 elliptical slots
B. The Slot Width
The width of each elliptical slot, which is 2times the minor
radius of the ellipse, is fixed at 5mm. This is calculated as
follows: for X-band SWAs, the width of a rectangular slot
the mostly used in the literature is 0.062500, corresponding to
a= 0.900. By proportionality, the width of the elliptical slot
for this S-band SWA is computed as follows:
Sl otW idth =a×0.0625
0.9= 2.84×0.0625
0.9= 0.19700 = 5mm
C. The Slot Displacement
By slot displacement, we refer to the distance between the
center of a slot and the centerline of the waveguide broadface,
as illustrated in Fig. 2.
With uniform slot displacement, all slots are at the same
distance from the centerline. This is similar to the case of
antenna arrays with discrete elements having equal excitation,
which results in an SLL around -13 dB. Lower SLLs are
obtained upon using non-uniform slot displacements. In both
the uniform and non-uniform displacement cases, the slots
should be placed around the centerline in an alternating order.
This is done to ensure the high efficiency of the antenna.
Figure 2. Slotted waveguide with 10 elliptical slots
The value of the uniform slot displacement that leads to a
good reflection coefficient is given by [11], [12]:
du=a
πsarcsin 1
N×G,(2)
where
G= 2.09 ×a
b×λg
λ0
×cos(0.464π×λ0
λg
)−cos(0.464π)2
.
(3)
In Equation 2, Nis the number of slots, which is equal to
10. In 3, λ0= 100 mm at 3 GHz. Combining 2 and 3, duis
found to be 7.7mm.
D. The Slot Length
For rectangular slots, the length is usually 0.98 ×λ0/2'
λ0/2. Because of the narrower ends of elliptical slots, their
length (double the major radius) is expected to be slightly
larger than λ0/2. The optimized elliptical slot length is
determined as follows: the SWA is modeled assuming a
uniform displacement (du= 7.7mm) and an initial length
of 0.98 ×λ0/2; the length is increased while watching the
computed reflection coefficient S11 until the antenna resonates
at 3 GHz with a low S11 value. In our case, the elliptical slot
length is found to be 54.25 mm.
For these uniform displacement and slot length, the resulting
sidelobe level ration (SLR) is around 13 dB, which is as
expected. The reflection coefficient S11 and the YZ-plane
gain pattern in this case are given in Figs. 3(a) and 3(b),
respectively. A peak gain of 16.6dB and an SLR of 13.2
dB are recorded. The half-power beamwidth (HPBW) in this
plane is 7.2degrees. These values are obtained using CST
Microwave Studio, and then verified with ANSYS HFSS.

(a) S11 for uniform slots displacement
(b) Pattern in the YZ plane
Figure 3. Antenna’s reflection coefficient and YZ-plane pattern for the case
of uniform slot displacement
III. NON -UNIFORM DIS PL ACE ME NT CA LC UL ATIO N
PROC ED UR ES
Our simulations have proven that the resonating length of
the elliptical slots is not very sensitive to the slots displacement
from the center line. For this, in the next calculations the length
of all slots is fixed at 54.25 mm. The displacement of the nth
slot is related to its normalized conductance gnby [11], [13]–
[15]:
dn=a
πarcsin v
u
u
u
t
gn
2.09λg
λ0
a
bcos2πλ0
2λg,(4)
gn=cn
N
P
n=1
cn
.(5)
In 5, Nis the number of slots, and cnsare the distribution
coefficients that we should determine to end up with the
desired SLL. Equation 5 guarantees that
N
P
n=1
gn= 1.
Several distributions (tapers) well-known in discrete antenna
arrays can be used to generate the cns(e.g. Taylor, Cheby-
shev). However, the resulting SLL of the SWA is always higher
than the SLL used for the discrete array distribution. To reach
the desired SWA SLL, a few iterations of the simulation setup
are required, where in each the SLL of the discrete array used
to generate the taper values is decreased. Illustrative examples
are shown below to further highlight this design procedure.
A. Example 1: 20 dB SLR with Chebyshev Distribution
In this example, our target is an SLR of 20 dB, where the
cnsare selected according to a Chebyshev distribution.
1) Coefficients and Slots Displacements: The coefficients
cnsfor a Chebyshev distribution are calculated from equations
in [16], [17]. For a −35dB Chebyshev taper, the cnsand their
corresponding slots displacements, calculated from Equation
4, are given in Table I. The −35dB Chebyshev taper has been
selected after some simulation iterations, as it provides the
desired −20dB SLL for the SWA. A −20dB Chebyshev taper
leads to SWA sidelobes higher than the −20dB goal.
TABLE I
TABL E:−35DB C HEB YS HEV TA PER CO EFFI CIE NTS A ND
CORRESPONDING SLOT DI SPL ACE ME NTS LE AD ING T O AN SWA SLL OF
-20 DB
Slot number Chebyshev coefficient Displacement (mm)
1 1 3.74
2 2.086 5.42
3 3.552 7.11
4 4.896 8.4
5 5.707 9.11
6 5.707 9.11
7 4.896 8.4
8 3.552 7.11
9 2.086 5.42
10 1 3.74
It is clear from Table I that the slots near the two waveguide
ends are closest to the broadface center line, whereas those
toward the waveguide center have the largest displacement.
This property applies to all the examples.
2) Results: For the previously determined slot parameters
(length, width, and coordinates), the SWA computed results
show a resonance at 3 GHz, an SLR of 20 dB, and a peak
gain of 16.1dB. The YZ-plane HPBW has increased to 8.4
degrees, compared to the uniform displacement case, as shown
in Fig. 4. The broadening of the main beam is expected since
the sidelobes have been forced to go lower.
Figure 4. Antenna’s YZ-plane pattern for the case of non-uniform slot
displacement with Chebychev distribution. A 20dB SLR is obtained.

Figure 5. Antenna’s YZ-plane pattern for the case of non-uniform slot
displacement with Taylor (one-parameter) distribution for an SLR of 20dB
B. Example 2: 20 dB SLR with Taylor (One-Parameter) Dis-
tribution
In this example, the SWA is designed to have an SLR of
20 dB, where the cnswill be obtained from a Taylor one-
parameter distribution [18].
1) Coefficients and Slots Displacements: The cnsfor a
Taylor one-parameter distribution can be computed using the
equations in [19] or [20]. For a −20dB SLL for the SWA, a
−30dB Taylor (one-parameter) taper is required. The resulting
coefficients, and the corresponding slots displacements are
listed in Table II.
TABLE II
TABL E:−30DB TAYL OR (O NE- PARA ME TER ) CO EFFIC IE NTS A ND
CORRESPONDING SLOT DI SPL ACE ME NTS LE AD ING T O AN SWA SLL OF
-20 DB
Slot number Taylor-based coefficient Displacement (mm)
1 1 3.493
2 2.467 5.518
3 4.137 7.194
4 5.597 8.419
5 6.449 9.070
6 6.449 9.070
7 5.597 8.419
8 4.137 7.194
9 2.467 5.518
10 1 3.493
2) Results: For the slots displacements in Table II, the
antenna keeps its resonance at 3GHz, shows an SLR of about
20 dB, and has a peak gain of 16 dB. The YZ-plane HPBW is
8.5 degrees, as shown in Fig. 5. It is to note that for the same
SLL of −20dB, the determined Chebyshev and Taylor (one-
parameter) coefficients have led to almost identical radiation
patterns, HPBW and gain parameters.
C. Example 3: 30 dB SLR with Taylor (One-Parameter) Dis-
tribution
In this example, we use a Taylor (one-parameter) distri-
bution to obtain an SWA SLR of 30dB. For this, the taper
coefficients for a −40dB Taylor (one-parameter) distribution
are required, and these are listed in Table III alongside
Figure 6. Antenna’s YZ-plane pattern for the case of non-uniform slot
displacement with Taylor Taylor Line-Source distribution for an SLR of 30dB
their corresponding slots displacements. The results show an
antenna resonance at 3 GHz, and a peak gain of 15.3dB. The
−30dB SLL has been attained, and the YZ-plane HPBW has
increased to 10 degrees. The YZ-plane gain pattern is shown
in Fig. 6.
TABLE III
TABL E:−40DB TAYL OR (O NE- PARAMETER) COE FFIC IEN TS A ND
CORRESPONDING SLOT DI SPL ACE ME NTS LE AD ING T O AN SWA SLL OF
-30 DB
Slot number Taylor-based coefficient Displacement (mm)
1 1 1.631
2 6.611 4.215
3 16.828 6.785
4 28.573 8.937
5 36.519 10.181
6 36.519 10.181
7 28.573 8.937
8 16.828 6.785
9 6.611 4.215
10 1 1.631
D. Example 4: Binomial Excitation
Although it is not directly possible to use a binomial
distribution to control the SLL, it is interesting to use the cns
from a binomial distribution and observe the resulting SWA
SLL. The binomial coefficients are obtained from the binomial
expansion:
(1 + x)m−1= 1 + (m−1)x+(m−1)(m−2)
2! x2
(m−1)(m−2)(m−3)
3! x3+... (6)
For the case of 10 slots, the binomial coefficients and the
resulting slots displacements are listed in Table IV. For these
displacements values, the obtained SLR is 38.6dB, and the
peak gain is 15 dB. The YZ-plane HPBW increases to 11.1
degrees, as shown in Fig. 7.

TABLE IV
TABL E: BINOMIAL CO EFFI CIE NT S AND CORRESPONDING SL OT
DIS PLAC EM ENT S
Slot number Binomial coefficient Displacement (mm)
1 1 0.964
2 9 2.900
3 36 5.847
4 84 9.069
5 126 11.268
6 126 11.268
7 84 9.069
8 36 5.847
9 9 2.900
10 1 0.964
Figure 7. Antenna’s YZ-plane pattern for the case of non-uniform slot
displacement with binomial distribution
E. Reflection Coefficients
For all the above examples, the SWAs retain their resonance
at 3GHz. The S11 plots for the four examples are shown in
Fig. 8.
-25
-20
-15
-10
-5
0
2.6 2.8 3 3.2 3.4
Reflection Coefficient [dB]
Frequency [GHz]
Chebyshev with -35 dB
Taylor with -30 dB
Taylor with -40 dB
Binomial
Figure 8. Reflection coefficient plots for the four illustrated examples
IV. CONCLUSION
This paper presented a simple procedure for designing
SWAs with specified SLLs. General guidelines for the slots
width, length and longitudinal positions are first given. The
offsets of the slots positions with respect to the waveguide
centerline, which determine the SLL, are then obtained from
well-known distributions. An intuitive rule regarding the used
distribution can be deduced, which is to select a distribution
with an SLL 15 dB lower than the desired SWA SLL. Il-
lustrative examples showing the distribution coefficients, slots
displacements, resulting patterns and S11 plots were given.
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