# Global Synthesis Method for the Optimization of   Multifeed EBG Antennas

Authors: Julien Drouet; Marc Thevenot; Régis Chantalat; Cyrille Menudier; Majed Koubeissi; Thierry Monédière; Bernard Jecko
Journal: International Journal of Antennas and Propagation
 (2008)

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## Abstract

This paper presents a novel technique for synthesizing a given radiation pattern from an EBG antenna with an array feed. The method determines the optimum sets of input waves and input impedances for the feed ports in order to perform simultaneously the radiation pattern and the impedance matching of all the radiating probes that form the array feed. The method is validated through a numerical design of an EBG antenna excited with four patch antennas. The structure is designed to radiate with a single lobe scanned atθ=30∘ in the E-plane. The interactions between each patch inside the EBG resonator are characterized with the CST MWS software. The optimum weights and the input impedances which simultaneously perform the objective radiation and the matching of all feeding ports are calculated by the developed global synthesis method. The feed network is designed with the Agilent ADS software in order to perform the specified weights and the impedances matching.


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## Body

## 1. Introduction

The electromagnetic band gap (EBG) antennas have
been extensively studied over the last decade. Their functioning principle is
as the leaky wave antennas [1–4].  A partially
reflective wall (PRW), an EBG material [5–7], or a
frequency selective surface (FSS) [8–11], for instance,
lies above a metallic ground plane in order to build a planar waveguide. The
first electromagnetic waveguide mode can radiate through the PRW, while its
phase velocity is greater than the light’s one, in other words, when the
operating frequency is close to the cutoff of the waveguide mode. This leaky
mode is excited with a radiating feed (like a patch antenna) or a set of
several feeds (like an array of small antennas [12–17]). The
resulting radiation pattern of the whole EBG structure depends on the radiation
of the feeds. The excitation of the EBG antenna with a feed array is useful for
the next two reasons:(i)
several feeds can be distributed under the PRW to enhance the
directivity without reducing the bandwidth [12]; the distance between the feeds
can be enlarged beyond one half wavelength because the EBG material is
efficient to degrade the grating lobes [13]; thus, it is relatively easy to
reach high directivities with few probes; this can simplify the design of the
feed network;(ii)
an array antenna with few feeds can be used to select a steered
angle; the radiations of the different feeds under the EBG PRW are reinforced
in a desired direction and suppressed in undesired directions, while the EBG
structure purpose is to enhance the gain in this desired direction [14–17].
The functioning of an array feed for an EBG
resonator seems different from the functioning of an usual phased array antenna
[18–21]. Since the
radiating probes lie between the ground plane and the PRW, the waveguide mode
interacts strongly with them. Consequently, their input impedances are
different from the initial impedances found without the PRW, the mutual
couplings could be ten times greater than without the PRW. Moreover, their
radiation patterns are strongly conditioned by the interactions with both the
EBG resonator and the others nearby probes. So, if no cares are taken when
designing, the strong mutual couplings between the probes can damage the
functioning of the feed network.Thus, a rigorous technique must be used to design
the array feed for the EBG antennas. It is described in Section2 of the paper.
The aim is to realize simultaneously an objective radiation pattern and the
greater radiation efficiency for a system made up of a multifeed EBG antenna and
its feed network.In Section3, the technique is illustrated
through the numerical studies of an EBG structure excited by four patch
antennas. The whole EBG antenna has been designed to radiate in the direction θ=30° in the E-plane [4].


## 2. Description of the Global Synthesis Method

This section
explains the method that we have used for designing a multifeed EBG antenna made
up of an array feed and a feed network. The problem is the optimization of this
feed network by taking into account the strong couplings between the output ports
of the array feed. The design technique is broken down into four steps. (1) The method
requires a full-wave analysis (e.g., CST Microwave studio) to characterize the
interactions between each probe inside the EBG resonator. During this analysis,
each radiating probe is successively supplied in order to extract the antenna'sS matrix and the radiation pattern of each probe. (2) The objective radiation
pattern is synthesized by a linear combination of the probes’ radiation
patterns. (3) The optimum input impedances and the weights which simultaneously
perform the objective radiation and the matching of all the probes are
calculated by taking into account the strong interactions between the radiating
probes. (4) Finally, the feed network can be designed to meet these
specifications.


### 2.1. Full-Wave Analysis of the EBG Antenna

This first step is a full-wave
analysis of the EBG antenna with the array feed. During this analysis, each
probe of the array feed is connected to an individual port in order to
characterize the interactions between the probes and the EBG structure. The
impedances of the port are annotatedZg. The probes are successively
supplied with a normalized incident wave a in order to extract their own
radiation pattern called “active element pattern” [20] and the scattering
matrix [Sc]. The active element patterns (Φi) are
vectors containing the electromagnetic fields (Ei(θ×φ)).


### 2.2. Synthesis of the Objective Radiation Pattern

The objective radiation pattern
must be fit by a linear combination of the different “active element patterns.”
This second step consists to find the complex coefficients (or weight) for the
linear combination. The objective radiation pattern (Φobj)
can efficiently be formed by combining an array factor with the transfer
function of the EBG material [15–17]. This linear
combination is given by (1). The coefficients β1,…,N are the elements of the
weighting vector.  A common matrix left division (computed by Gaussian
elimination) can be used to deduce the best combination of the N radiation
patterns (the β vector): (1)[Φ1Φ2⋯ΦN]︸(θ*φ;N)⋅(β1β2⋮βN)≈(Φobj)︸(θ*φ).


### 2.3. Definition of the Specifications for  the Feed Network

The global synthesis method
provides the specifications for the feeding ports. The aim is to optimize the radiation
efficiency while maintaining the objective radiation function. These
specifications are the normalized incident wave and the input impedance that
must be connected to each port of the multifeed EBG antenna.Figure1 is a schematic
representation of one radiating probe of the array feed. It is connected to a Zgi impedance port. In this figure, Zei is the intrinsic impedance of
the probe, and Imi is a current resulting from the mutual coupling
with the other radiating probes. The current and the voltage in each probe are evaluated.
According to the superposition theorem, the driven voltages
ui
, which must excite each probe, are derived from (2). Let us notice that this equation
takes into account the mutual coupling between the probes: (2)(u)=(diag(Zg)⋅[I+Sc]⋅diag(a))⋅(β). Vector (a)
and vector (Zg) are, respectively, the waves that are used to supply each probe
during the electromagnetic analysis of the EBG structure (first step) and the
impedances of the ports.Figure 1
Antenna “i” connected to a
port of impedance Zgi.The  currents
into the ports are deduced from (3):
(3)(i)=[α]⋅(u),(4)[α]=[diag(1Zg)⋅[I−Sc]⋅diag(a)]⋅⋯[diag(Zg)⋅[I+Sc]⋅diag(a)]−1.As for the
currents, the input impedances depend on both the coupling matrix Sc and the voltages
that are needed to supply the probes. They are the elements of a scan impedance
matrix(Zscan) which is given by (6):
(5)Zscan=u⋅/([α]⋅u). According
to (3) and (4), (5) becomes (6)Zscan=([diag(Zg)⋅[I+Sc]⋅diag(a)]⋅β)⋅/⋯([diag(1/Zg)⋅[I−Sc]⋅diag(a)]⋅β).


### 2.4. Design of the Feed Network

In order to
maximize the radiation efficiency, the ports of the feed network must match theZscan impedances. Each port “i” of the feed network must
excite the ports of the EBG antenna with the incident waves specified by the
relation (7) because the Zscan impedances are achieved only when the
relation (2) is fulfilled: (7)ai=uiRe(Zscan)Zscan.When these conditions (6), (7) will
be fulfilled by the feed ports, the coupled power does not disturb
the functioning of the feed network. So, the whole incident power is
transmitted to the probes, the radiation efficiency is maximized and the weighting
vector (β) is realized.


## 2.1. Full-Wave Analysis of the EBG Antenna

This first step is a full-wave
analysis of the EBG antenna with the array feed. During this analysis, each
probe of the array feed is connected to an individual port in order to
characterize the interactions between the probes and the EBG structure. The
impedances of the port are annotatedZg. The probes are successively
supplied with a normalized incident wave a in order to extract their own
radiation pattern called “active element pattern” [20] and the scattering
matrix [Sc]. The active element patterns (Φi) are
vectors containing the electromagnetic fields (Ei(θ×φ)).


## 2.2. Synthesis of the Objective Radiation Pattern

The objective radiation pattern
must be fit by a linear combination of the different “active element patterns.”
This second step consists to find the complex coefficients (or weight) for the
linear combination. The objective radiation pattern (Φobj)
can efficiently be formed by combining an array factor with the transfer
function of the EBG material [15–17]. This linear
combination is given by (1). The coefficients β1,…,N are the elements of the
weighting vector.  A common matrix left division (computed by Gaussian
elimination) can be used to deduce the best combination of the N radiation
patterns (the β vector): (1)[Φ1Φ2⋯ΦN]︸(θ*φ;N)⋅(β1β2⋮βN)≈(Φobj)︸(θ*φ).


## 2.3. Definition of the Specifications for  the Feed Network

The global synthesis method
provides the specifications for the feeding ports. The aim is to optimize the radiation
efficiency while maintaining the objective radiation function. These
specifications are the normalized incident wave and the input impedance that
must be connected to each port of the multifeed EBG antenna.Figure1 is a schematic
representation of one radiating probe of the array feed. It is connected to a Zgi impedance port. In this figure, Zei is the intrinsic impedance of
the probe, and Imi is a current resulting from the mutual coupling
with the other radiating probes. The current and the voltage in each probe are evaluated.
According to the superposition theorem, the driven voltages
ui
, which must excite each probe, are derived from (2). Let us notice that this equation
takes into account the mutual coupling between the probes: (2)(u)=(diag(Zg)⋅[I+Sc]⋅diag(a))⋅(β). Vector (a)
and vector (Zg) are, respectively, the waves that are used to supply each probe
during the electromagnetic analysis of the EBG structure (first step) and the
impedances of the ports.Figure 1
Antenna “i” connected to a
port of impedance Zgi.The  currents
into the ports are deduced from (3):
(3)(i)=[α]⋅(u),(4)[α]=[diag(1Zg)⋅[I−Sc]⋅diag(a)]⋅⋯[diag(Zg)⋅[I+Sc]⋅diag(a)]−1.As for the
currents, the input impedances depend on both the coupling matrix Sc and the voltages
that are needed to supply the probes. They are the elements of a scan impedance
matrix(Zscan) which is given by (6):
(5)Zscan=u⋅/([α]⋅u). According
to (3) and (4), (5) becomes (6)Zscan=([diag(Zg)⋅[I+Sc]⋅diag(a)]⋅β)⋅/⋯([diag(1/Zg)⋅[I−Sc]⋅diag(a)]⋅β).


## 2.4. Design of the Feed Network

In order to
maximize the radiation efficiency, the ports of the feed network must match theZscan impedances. Each port “i” of the feed network must
excite the ports of the EBG antenna with the incident waves specified by the
relation (7) because the Zscan impedances are achieved only when the
relation (2) is fulfilled: (7)ai=uiRe(Zscan)Zscan.When these conditions (6), (7) will
be fulfilled by the feed ports, the coupled power does not disturb
the functioning of the feed network. So, the whole incident power is
transmitted to the probes, the radiation efficiency is maximized and the weighting
vector (β) is realized.


## 3. Numerical Validation

In this section, a proof is
given to how useful and efficient the global synthesis method is when designing
EBG antennas. For this purpose, the following studies will focus on the design
of a leaky wave EBG antenna which is excited by four
radiating patches.The PRW is a 2.466 mm thick alumina slab with a
permittivity of 9.4. It lies 15.0 mm above the ground plane. The EBG structure
is designed to enhance the directivity in the directionθ=30° at the frequency f0=11.35 GHz
[4, 15–17]. The global EBG
antenna dimensions are Lx=330 mm and Ly=280 mm. The antenna is widely oversized
in order to neglect the edge effects. The four patches are organized into a 2×2
array (Figure 2) and spaced out by 13.2 mm (0.5λ0). They are printed on a 0.5 mm thick Teflon (εr=2.08) substrate. The microstrip feed network which
has to generate the weights and the complex conjugate of the patch impedances must
not interfere with the antenna. We have chosen to print it behind the antenna ground
plane onto another Teflon substrate. Four 50 Ω coaxial transitions are drilled through the
ground plane and used to connect the microstrip ports to the patch antennas
(Figure 3).Figure 2
The EBG leaky wave
antenna is fed with a patch array. It must radiate in the direction[θ=30°;φ=0°].Figure 3
The patches are printed onto a Teflon substrate. They are
fed through the ground plane by a 50 Ω coaxial probe. A second Teflon substrate is
set onto the back face of the metallic plane. It receives the printed feed
network.An evaluation of the ability
for the EBG antenna to radiate with a single lobe in the directionθ=30° was first made with the
analytical method described in [4, 8, 15–17]. This
analysis provides a theoretical radiation pattern that can constitute the
objective one (Φobj). The EBG spatial filter
enhances the radiations nearly the θ=30° direction whatever the φ angle [4, 14–17].  The patch array is then weighted in order to
eliminate the radiation in the opposite direction ({θ=30°;φ=180°} or  {θ=−30°;φ=0°}).  The
radiation pattern of the multifeed EBG antenna is estimated by multiplying the
EBG spatial filtering function with the patch array radiation pattern. The
patch array radiation pattern, the spatial filtering function, and their product
are plotted in Figure 4. The weights, which are applied to the patches in order
to obtain the objective radiation, are written in Table 1. Only two weights
appear in the table because the structure is symmetric. It should to be
stressed that these results are approximated since the analysis considers an
infinite dimensional EBG structure and the patch antennas do not interfere with
each other. The radiation pattern illustrated by the black solid line in Figure
4 can be used as the objective radiation pattern Φobj (1).Table 1
Theoretical array factor.
WeightsPatch 1Patch  2Magnitude11Phase0°90°Figure 4
The patch array (2×2 patches) selects a radiation direction within
the conical beam steering.In the next step, we have used CST
Microwave studio to achieve the full-wave analysis of the whole EBG structure. Thanks to the symmetry, only two patches among the four are supplied
consecutively in order to obtain their own radiation patterns.
These “active element patterns” are plotted in Figure5. According to (1), the
weights β1 and β2 are deduced and written in Table
2. Thus, Figure 6 points out the resemblance between the objective radiation
pattern and the linear combination of the active element patterns weighted by
the coefficients β1 and β2.Table 2
Weight that may be assigned to the
radiation patterns of active element at 11.35 GHz.
ββ1 (patch 1)β2 (patch 2)Magnitude10.585Phase0°73°Figure 5
Active element patterns [20] when the EBG structure is successively fed by the patches 1 and 2 (E-plane: 11.35 GHz).Figure 6
Comparison of the objective radiation pattern with the linear combination of the active element patterns (E-Plane: 11.35 GHz).Figure7 shows
the scattering matrix of the patch array with (right) and without (left) the
PRW above the ground plane. The comparison shows that the mutual coupling
between the patches is dramatically increased (about 10 dB) by the interactions
with the EBG structure. These interactions should not be omitted when
connecting the array feed with the feed network.S parameters of the
array antenna (2×2 patches) and S parameters of the same array
antenna inside the EBG structure.
(a)(b)The optimum weights and the
input impedances which simultaneously perform the objective radiation and the
matching of all the feeding ports can be calculated. The driven
voltages(u1,u2), the impedances (Zscan1,Zscan2), and the weights (a1,a2) are obtained
from (5), (6), and (7), the scattering matrix and (β1,β2). They are shown in Table 3. Even
though the patch antennas seem well matched to 50 Ω when they are
successively fed (Figure 7), it appears that the impedances that optimize the
efficiency are strongly affected by the mutual couplings. They are quite
different from 50 Ω.Table 3
Electrical characteristics of the patches array,F=11.35 GHz.
WeightsPatch 1Patch  2VoltageUiMagnitude10.7263Phase0°50.3°ImpedancesZscan50.7+25.6 jΩ83.7−47.5 jΩaiMagnitude0.8760.482Phase0°106°The design of the microstrip feed network
has been made with the Agilent ADS (Momentum) software in
order to perform the weights and the impedance matching specified by the global
synthesis method (Table3). When designing the feed network, the
impedances of the ports are set equal to the impedances Zscan. Due
to the magnetic symmetry, only one arm of the network is shown (Figure 8).Figure 8
The feed network is matched to impedancesZscan_1/2 and it achieves the weights shown in Table 3.We consider
two validations for the global design with CST. First, the scattering matrix of
the feed network and the EBG antenna are numerically connected together. The
radiation pattern and the return loss are computed. These results are then
compared with an entire structure simulation. For this entire full-wave
simulation, the feed network and the EBG antenna are meshed under an unique
computational volume. Figure9 plots the radiation patterns in the E-plane. The
two validations (entire structure simulation and the S matrix connection) are
compared with the objective linear combination of the active element patterns (β1⋅ϕ1(θ,φ)+β2⋅ϕ2(θ,φ)).
It is obvious that the two validations agree well with the objective radiation
pattern. We can notice a minor difference in the backward directions for the
entire structure simulation. It is probably caused by the direct radiation of
the feed network. The return loss is plotted in Figure 10. The level is lower
than −25 dB at the optimization frequency (11.35 GHz) for the two validations and
it stay lower than −15 dB over a wide-frequency band.Figure 9
Radiation pattern comparison in the E-plane. 3 radiation patterns are
plotted. The first plot is obtained by the linear combination of the active
element patterns; the second is CST MWS posttreatment taking into account the
feed network scattering matrix; and the third plot results from an entire
structure simulation (the feed network and the EBG antenna are meshed under the
same CST simulation).Figure 10
Return loss of the antenna. S11 is calculated by connecting the antenna with the feed network
scattering matrix (grey bold line). This S11 is compared with the S11 computed
when the antenna and its feed network are meshed under the same CST MWS
simulation (black line).The numerical validations show
that the radiation pattern is successfully synthesized as well as the impedance
matching of every port. Through the
precision of our results, we demonstrate that the method is very efficient for
synthesizing a given radiation pattern from an EBG antenna with an array feed.


## 4. Conclusion

This paper
illustrates the usefulness and the efficiency of using a global synthesis
method when designing an array feed for an EBG antenna. In such structures, the
radiating probes strongly interact with each other and with the EBG material.
The intensity of the coupled waves may damage the functioning of a feed network
if no cares are taken. The global synthesis method provides the input
impedances and the weights for the input waves that optimize the efficiency of
the antenna, while an objective radiation pattern is accomplished. To sum up, the
feed network can simply be designed with momentum, while the used port
impedances are taken as the optimization impedances of the array feed for the
EBG antenna.



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*Source: 790358-2008-10-09.xml*