Antenna design

Using the cavity model, the resonant frequencies of the TM mode of a circular patch are given in equation (1) [Reference Balanis23]:

(1)$$f_r = {cX_{np}\over 2\pi a_{eff}\sqrt {\epsilon_r}}\comma\; $$

where $c$ is the light velocity in the free space, $\epsilon _r$ is the relative permittivity of the substrate, and $X_{np}$ are the zeros of the derivative of the Bessel function ${\rm Jn}\lpar x\rpar$ of order $n$, as is true of TE-mode circular waveguides. The lowest-order mode, ${\rm TM}_{11}$, uses $X_{11} = 1.84118$ and produces a linearly polarized field similar to a square patch. $a_{eff}$ is an effective radius of the patch taking fringing into account. As is known, the fringing gives a wider electrical aspect to the patch, and it has been taken into account by introducing a correction factor. For the circular patch, a correction is introduced using an effective radius $a_{eff}$, to replace the real radius $a$, given in [Reference Balanis23]:

(2)$$a_{eff} = a\left\lbrace 1 + {2h\over \pi \epsilon_r a}\left(\ln\left({\pi a\over 2h}\right) + 1.7726\right)\right\rbrace^{1/2}\comma\; $$

where $h$ is the height of the substrate. Since the effective and physical radii of the patch are nearly the same, equation (2) may be iterated to compute the real radius $a$ given by equation (3):

(3)$$a = {F\over \left\lbrace 1 + \frac{2h}{\pi \epsilon_{rF}}\left(\ln\left(\frac{\pi F}{2h}\right) + 1.7726\right)\right\rbrace^{1/2}}$$

where $h$ is in cm, and

(4)$$F = {8.791\times 10^{9}\over f_r\sqrt {\epsilon_r}}$$

By relating the areas of the circular and hexagonal patches, equation (1) can be applied for designing an HMPA. Let there be a circular patch of radius $a$; the area of this patch is:

(5)$$area_{circle} = \pi a^{2}$$

A regular hexagon is constructed by drawing six equilateral triangles. As is known the area of an equilateral triangle of side $s$ is given by equation (6):

(6)$$area_{equi{\textrm -}triangle} = {\sqrt{3}s^{2}\over 4}$$

Then the area of a regular hexagon is given by equation (7):

(7)$$area_{regul{\textrm -}hexagon} = {3\sqrt{3}s^{2}\over 2}$$

The hexagon side is determined by comparing the areas of hexagonal and circular patches as shown in equation (8):

(8)$$\pi a_{eff}^{2} = {3\sqrt{3}s^{2}\over 2}$$

By taking fringing into account, the hexagonal side is:

(9)$$s = 1.0997 a_{eff}$$

Cantor set fractal theory

Cantor dust described by mathematician Georg Cantor in 1872 is probably the oldest known fractal and the easiest to create [Reference Ali, Abdulkareem, Hammoodi, Salim, Yassen, Rashed Hussan and Al-Rizzo24–Reference Reha, El Amri, Benhmammouch, Said, El Ouadih and Bouchouirbat26]. The construction of this structure is based on a straight line segment from which the central third is removed. The same operation is performed on the two remaining segments, then by successive iteration on the resulting smaller and smaller segments. This shape is characterized by a number of segments tending to become infinite with an almost zero length. The number of copies of the original shape obtained from one iteration to another is equal to 2 ($N = 2$) and the size of each new copy is equal to 1/3 of the original size ($r = 1/3$). This geometry is described by using the iterated function system (IFS) which is represented by the following affine transformation [Reference Rusu, Baican and Minin10]:

(10)$$w_i \left[\matrix{x\cr y }\right] = \left[\matrix{a_i & b_i \cr c_i & d_i }\right]\times \left[\matrix{x \cr y }\right] + \left[\matrix{e_i \cr fi }\right]$$

The IFS coefficients for the Cantor set are given in Table 1. In general, the fractal geometry has a very particular feature that its dimension exceeds its topological dimension, indeed there are various types of representations used for the fractal dimensions such as Hausdorff dimension defined as:

(11)$$D = {\ln\lpar n\rpar \over \ln\lpar r\rpar }\comma\; $$

where $r$ is the scaling factor and $N$ is the number of self-similar structure copies. The Cantor set fractal consists of $N = 2$ congruent subsets, where each gives the original set when it is enlarged by a factor $r = 3$. Therefore, the fractal dimension of the Cantor set is defined as in equation (12):

(12)$$D = {\ln\lpar 2^{n}\rpar \over \ln\lpar 3^{n}\rpar } = {\ln\lpar 2\rpar \over \ln\lpar 3\rpar } = 0.630929 = 0.631\DIFadd{.}$$

Table 1.

IFS transformation coefficient for Cantor set fractal

The construction of the investigated antenna is started with a simple hexagonal patch. The antenna is manufactured on an FR4 epoxy substrate with an $\epsilon _r$ of 4.4 and a loss tangent of 0.0028. The volume of the antenna is $Ls \times Ws \times h = 31 \times 28 \times 1.6\, {\rm mm}^{3}$, with a partial ground plane of $Lg \times Ws$ at the bottom of the substrate (Fig. 1). The $50\, \Omega$ impedance is achieved by adjusting the feed line width and length (respectively $wf$ and $Lf$), the gap ($g$) notch width, and the inset distance ($d$) from the radiating edge. The antenna was designed and simulated using CST Microwave Studio, which uses the finite integration technique for computation. To show the effect of inserting fractal geometry into the patch, the structure has been designed with three iterations of the Cantor set fractal geometry that has been introduced on the hexagonal patch. The side $s$ of the hexagon shape is computed using equation (9). At a resonant frequency, the evaluated hexagonal side $s$ is of 21.92 mm.

Fig. 1.

The proposed antenna configuration: (a) front view and (b) back view.

Parametric study

The antenna performance is affected by several parameters such as the distance ($d$) between the radiating patch and feed line, the length ($Lg$) of the ground plane, the length ($Lf$) and width ($wf$) of the feed line, the length ($Lc$) and width ($Wc$) of the Cantor set slots.

As shown in Fig. 2, changing the length of the ground plane $Lg$ not only changes the lateral dimensions of the antenna but also has a remarkable effect on the resonant characteristics of the proposed antenna. At the high $Lg$ values, the frequency performance degrades. The optimum ground plane length is $Lg = 11\, {\rm mm}$, for which the desired band frequency is obtained.

Fig. 2.

Effect of various lengths of the ground plane ($Lg$) on $S_{11}$.

To improve the attained characteristics, the proposed antenna is simulated by keeping the length $Lf = 12\, {\rm mm}$ and varying the width (wf) of the feed line. The return loss graphs of MPA along with various values of wf is shown in Fig. 3. It can be premeditated that the desired wideband characteristics have been exhibited with the feed line dimensions of $Lf = 12\, {\rm mm}$ and $wf = 3.5\, {\rm mm}$. The distance ($d$) between the radiation patch and the feed line has an important effect on the impedance matching of the antenna. From Figs 4 and 5, it can be elaborated that the desired features have been attained at the inset feed line distance of $d = 2\, {\rm mm}$ and a gap of $g = 1\, {\rm mm}$. The Cantor fractal slots are introduced in the upper part of the hexagonal patch. A simple parametric study of the width ($Wc$), the length ($Lc$) of these slots, and the space between them ($Espc$) have brought us back to results depicted in Fig. 6.

Fig. 3.

Effect of various widths of the feed line ($wf$) on $S_{11}$.

Fig. 4.

$S_{11}$ for various inset feed line distances ($d$).

Fig. 5.

$S_{11}$ for various gaps ($g$).

Fig. 6.

$S_{11}$ for various Cantor set dimensions values.

It can be claimed that the Cantor slots width ($Wc$) of 12 mm, length ($Lc$) of 1 mm, and ($Espc$) of 0.5 mm are found to be optimum, for which the desired DSRC frequency band of 5.9 GHz is achieved.

It may be noted from Fig. 7 that the structure with Cantor fractal slots presents an improved matching and good return losses compared with the one without the fractal slots. The hexagonal-shaped structure with a Cantor fractal geometry covers the desired band exhibiting good performances.

Fig. 7.

Return loss comparison of both configurations.