As mentioned before, CST is employed to model and simulate an eight-element AMAA for DVB-T broadcasting at 698 MHz. The feeding network is out of scope for this paper and therefore the array elements are considered to be directly excited at their input ports by sources. The inset feeding method is used in order to achieve better impedance matching to
sources. The geometry of AMAA is illustrated in
Figure 2 and is defined by the physical dimensions
W and
L of the rectangular array elements, four distances
between the array elements, where
(aperiodic structure), the transmission line length
and width
which connects each array element to its respective source, the inset depth
s and the inset width
g. PSOvm is employed to find the optimal values for all the above parameters. In order to help the optimization algorithm converge faster, the search space for all the geometry parameters must be confined between an upper and a lower limit. As indicated by Reference [
19] the theoretical values of
W and
L are given by:
where
c is the free space wave velocity,
is the operation frequency (698 MHz in this paper),
is the dielectric constant of the substrate used here,
is the effective dielectric constant, and
is the length reduction applied at both sides of the effective length to get physical length
L. It is noted that the subscript “th” denotes the theoretical value of a variable. The values of
and
are respectively estimated [
19] by:
where
h is the substrate height (or thickness). The substrate used in this paper is Duroid RT5880LZ with
and
mm [
20]. The thickness of the copper cladding (used in CST modeling) at both sides of the substrate is equal to 35
m [
20]. By applying the above equations, we get
mm and
mm. It is expected that the optimal values for
W and
L will not deviate more than
from the theoretical values given by (
1) and (
2) and thus,
W is limited between 0.6
and 1.4
while
L is limited between 0.6
and 1.4
. The distance
is confined between 0.5
and 0.8
(
is the free space wavelength at 698 MHz) where the maximum
is expected to be found. Since
,
,
and
are in incremental order to create the aperiodical structure of the array, we adopt the rule
where
are random numbers between 0 and 0.2
. Therefore, the parameters
,
and
are indirectly optimized through
,
and
, respectively. The values of
are confined between 0 and
which helps for impedance matching. As for the width of the transmission line
, the microwave theory predicts that a width of 8.4 mm will result in a transmission line of characteristic impedance equal to 50
however due to the mutual coupling between the array elements a more relaxed constraint needs to be applied in this variable and therefore the lower limit has been set at half the theoretical value (4.2 mm) and double (16.8 mm) for the upper limit. In order to calculate the theoretical value of
d we use the following equations [
19]:
where
is the self conductance of one of the radiating slots of a microstrip patch,
is the free space wavenumber,
is the mutual conductance between the two radiating slots of a microstrip patch,
is the Bessel function of the first kind of order zero and
is the resonant input resistance of a patch. By solving (
8) for
we obtain the theoretical value
mm and therefore
s is considered to be restricted between 0.6
and 1.4
. As mentioned earlier, eight feeding weights must be found in order to achieve the desired radiation pattern. The boundaries for the amplitudes of the feeding weights
(
) are set between 0.1 and 1 (considering an upper limit equal to ten times the lower limit), while the phases
are considered to be between
and
. The optimal value for
g according to Reference [
21] is considered to be between
and
. As
W takes different values during the optimization and
g relies on
W, it was decided to confine
g between
and
of the current value of
W (
) during the optimization process. All the boundaries for the optimization parameters are summarized in
Table 1. In total, there are 26 parameters to be optimized (
), thus creating a complex problem to solve.