Iterated Function System (IFS)

The IFS is a mathematical method to generate the fractal structure by describing scaling, rotation, and transformation of the kernel structure. It uses the set of self-affine transformations for fractal geometry generation. The IFS matrix of the proposed CSK fractal can be expressed as [Reference Sagan23]

(5)$$P\lpar {{x}^{\prime}\;{y}^{\prime}} \rpar = \left({\matrix{ a & b \cr c & d \cr } } \right)\left({\matrix{ {{x}^{\prime}} \cr {{y}^{\prime}} \cr } } \right)+ \left({\matrix{ e \cr f \cr } } \right)\comma \;$$

(6)$$P_1\lpar {{x}^{\prime}\;{y}^{\prime}} \rpar = \displaystyle{1 \over 2}\;\left({\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right)\left({\matrix{ {{x}^{\prime}} \cr {{y}^{\prime}} \cr } } \right)+ \displaystyle{1 \over 2}\;\left({\;\matrix{ 0 \cr 0 \cr } } \right)\comma \;$$

(7)$$P_2\lpar {{x}^{\prime}\;{y}^{\prime}} \rpar = \displaystyle{1 \over 2}\;\left({\matrix{ 0 & {-1} \cr 1 & 0 \cr } } \right)\left({\matrix{ {{x}^{\prime}} \cr {{y}^{\prime}} \cr } } \right)+ \displaystyle{1 \over 2}\;\left({\matrix{ 2 \cr 0 \cr } } \right)\comma \;$$

(8)$$P_3\lpar {{x}^{\prime}\;{y}^{\prime}} \rpar = \displaystyle{1 \over 2}\;\left({\matrix{ 0 & 1 \cr {-1} & 0 \cr } } \right)\left({\matrix{ {{x}^{\prime}} \cr {{y}^{\prime}} \cr } } \right)+ \displaystyle{1 \over 2}\;\left({\matrix{ 2 \cr 2 \cr } } \right)\comma \;$$

(9)$$P_4\lpar {{x}^{\prime}\;{y}^{\prime}} \rpar = \displaystyle{1 \over 2}\;\left({\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right)\left({\matrix{ {{x}^{\prime}} \cr {{y}^{\prime}} \cr } } \right)+ \displaystyle{1 \over 2}\;\left({\matrix{ 2 \cr 0 \cr } } \right).$$

Let us assume that P ₁, P ₂, … , P_n are the series of linear self-affine transformations and S be the initial structure. The final geometry is obtained by uniting all four transformations. It can be expressed as

(10)$$P\lpar S \rpar = \mathop \sum \limits_{n = 1}^N P_n \lpar S \rpar. $$

The higher-order iterations of the fractal are achieved by repeating this process. The IFS is also referred to as a multiple reduction copy machine. The fractal similarity dimension (D) can be calculated as [Reference Singh and Singh21]

(11)$$D = \displaystyle{{\log n} \over {\log [ {1/f} ] }}\comma \;$$

where n is the number of copies and f is the scaling factor. In the case of the Sierpinski Knopp curve, n = 4 and f = 1/2. The fractal dimension is 2 for the proposed CSK antenna.

Antenna configuration

The projected CSK fractal antenna is assembled on the FR4 dielectric material with a volume of 35 × 35 × 1.6 mm³. Figures 1(a) and 1(b) show the front and 3D view of the proposed CSK fractal antenna. The novelty of the proposed antenna lies in the antenna radiating structure. Also, a slot-loading technique is employed in order to increase the number of operating bands. The structure consists of the Sierpinski Knopp curve inspired square monopole antenna on the upper side of the substrate and partial ground plane on the flipside.

Fig. 1. Geometry of the proposed CSK fractal antenna. (a) Front view, (b) 3D view.

The microstrip feed line is bonded to the female SMA connector having 50Ω impedance for electrifying the antenna. The motivation of such fractal geometry is to increase the electrical length and bandwidth of the antenna. The variables are assigned to each element of the antenna for parametric analysis as follows: L_s – length of the substrate, W_s – width of the substrate, h – thickness of the substrate, L_p – length of the microstrip patch, W_p – width of the microstrip patch, L_g – Length of the ground plane, W_g – width of the ground plane, L_f – length of the microstrip feed line, and W_f – width of the microstrip feed line. The distance between the feed line and edge of the patch is denoted by W ₁.

To achieve the desired results, the dimensions of the substrate material, ground plane, microstrip patch, and feed line have been optimized by performing the parametric analysis. The performance metric of the antenna is investigated by utilizing the ANSYS HFSS. The values of the design variables are listed in Table 1, which yields the optimum results. The square of dimension 25 × 25 mm² has been chosen as a kernel to create the geometry of the projected fractal antenna. Initially, the square patch is fragmented into two isosceles triangles. The various iteration stages in the development of the proposed CSK fractal antenna from the basic design to 3^rd iteration is depicted in Fig. 2.

Fig. 2. Iteration Stages of fractal antenna. (a) Basic design, (b) 0th iteration, (c) 1st iteration, (d) 2nd iteration, (e) 3rd iteration.

Table 1. Dimensions of the CSK antenna

Figure 2(a) shows the square monopole antenna which is modified in such a way to achieve the desired frequency bands by using fractal geometry. As seen in Fig. 2(b), a square with an area of 8.83 × 8.83 mm² has been placed in the middle and rotated by 90°. The resulted diamond-shaped patch is etched from the basic square monopole, which is named as 0^th iteration. For the first iteration, the square monopole is sliced into four smaller squares. The diamond-shaped patch is scaled down by 2 and located in the middle of each four squares. The edges of all diamond-shaped patches are joined and removed from the kernel. Antenna design at the end of the first iteration is displayed in Fig. 2(c). This recursive process is repeated for obtaining the second and third iterations of the fractal antenna that are depicted in Figs 2(d) and 2(e), respectively.

Parametric study

The parametric analysis of the fractal antenna has been carried out for achieving better performance. In this study, the size of the substrate, ground plane, feed line, and square patch is considered for the parametric analysis. The S ₁₁ at different iteration levels from basic design to iteration-4 of the fractal antenna is shown in Figs 3(a) and 3(b). The antennas in the basic design and 0^th iteration have not an effective S ₁₁, which is higher than −10 dB. In the first iteration, antenna resonates at 6.28 GHz with −15.8 dB reflection coefficient.

Fig. 3. Effect on S ₁₁ at different iteration levels. (a) Basic design, Iteration-0, Iteration-1, Iteration-2 and Iteration-3, (b) Iteration-4.

The dual resonance appears at the second iteration where the S ₁₁ values are −15.02 dB at 2.42 GHz and −16.88 dB at 3.23 GHz. The required triple band with improved S ₁₁ and bandwidth is achieved by the suggested fractal antenna in the 3rd iteration. Reflection coefficients are reduced to −19.6, −24, and −17.3 dB at the corresponding resonant frequencies of 1.94, 4.95, and 6.41 GHz, respectively.

Table 2 presents the comparative characteristics of the antenna at various iterations. It is noted that the number of resonant frequency increases as the number of iterations increases because of the fractal structure.

Table 2. Comparative characteristics of antenna at various iterations

The bandwidth and gain of the antenna are decreased at some frequency bands and voltage standing wave ratio (VSWR) is acceptable (VSWR≤2) by increasing the iteration. When doing it up to the 3rd iteration, the performance such as bandwidth and gain is found to be better than the results obtained during 4th iteration. On the other hand, it is observed that moving from 3rd iteration to 4th iteration, the operating frequency is deviating and the antenna becomes quite complicated and its fabrication becomes difficult. Therefore, the number of iteration in the proposed work is limited to 3rd iteration.

The geometrical parameters such as L_S, W_S, L_P, W_P, L_g, L_F, W_F, and h are varied individually while the other parameters are remaining constant. The length (L_S) and width (W_S) of the substrate are changed from 31 to 35 mm. The slight change in the substrate length (L_S) alters the resonance behavior of the antenna. The S ₁₁ versus frequency for different L_S and Ws is shown in Figs 4(a) and 4(b), respectively. The required triple-band resonance characteristics with better performance are observed at L_S = W_S = 35 mm. From Fig. 4(b), it can be found that the frequency is shifted from 6 to 4.95 GHz due to changes in the substrate width (W_S).

Fig. 4. Effect on S ₁₁ with variation of (a) length of the substrate L_S, (b) width of the substrate W_S.

The patch length L_P and width W_P are varied from 21to 25 mm with a step of 1 mm. The impedance matching is found well at L_P = 25 mm.

As seen in Fig. 5(a), the antenna resonates at the higher frequencies above 6 GHz for the values of L_P from 21 to 24 mm. The antenna has dual-band operation for the values of W_P from 21 to 24 mm and triple-band operation at 25 mm which is presented in Fig. 5(b).

Fig. 5. Effect on S ₁₁ with variation of (a) length of the patch L_P, (b) width of the patch W_P.

In a further study, the feed line length (L_F) is varied from 5 to 9 mm with an interval of 1 mm and the effect on S ₁₁ is depicted in Fig. 6(a). The optimum value is chosen as 5 mm because of its resonance at 4.95 GHz, which is the required band for PPDR applications. The effect on S ₁₁ with respect to various feed width (W_F) is presented in Fig. 6(b). The resonant frequency is shifted to the lower side as W_F increases. It is noted that the antenna has good performance at 3 mm. The substrate thickness (h) also plays an important role in the antenna design. Figure 7(a) shows the impact on the S ₁₁ by varying h. A poor impedance matching is perceived for the values of h ranges from 1 to 1.6 mm. Antenna impedance is well matched at 1.6 mm. Therefore, the value of 1.6 mm is selected as the thickness of the substrate (h). Figure 7(b) shows the effect on S ₁₁ against frequency for different ground plane length (L_g). The better result is attained at 6.5 mm.

Fig. 6. Effect on S ₁₁ with variation of (a) length of the feed line L _F, (b) width of the feed line W _F.

Fig. 7. Effect on S ₁₁ with variation of (a) substrate thickness h, (b) length of the ground plane L_g.

All the geometrical parameters of the proposed CSK fractal antenna are chosen according to the parametric analysis for achieving optimum results.