II. MATHEMATICAL BACKGROUND OF X-SHAPED FRACTAL WITH MICROSTRIP FEED LINE

In this section, the design methodology of proposed X-shaped fractal geometry is discussed. Figure 1 shows the design process of the proposed fractal shape. Initially at stage-1, it consists of two perpendicular metal strips both having length and width of L ₁ and W respectively. The width of all metal strips remains unchanged in all stages. In stage-2, the length of the two perpendicular strips (L ₂) is m × L ₁, where m is the scale factor whose value lies between 0 and 1. Hence, L ₂ is smaller than L ₁. For stage-2, four pairs of these X-shaped perpendicular strips of length L ₂ and width W are then added on all four corners of X of stage-1. The same procedure is applied in next two stages (i.e. third and fourth stage). Finally, the proposed X-shape fractal is achieved at stage-4 as shown in Fig. 2. The length of perpendicular strips of i _th stage can be determined by the following expression:

(1) $$L(i + 1) = m\; L(i)\quad (i = 1,\; 2,\; 3),$$

where, m is multiplying factor and i is iteration number and L ₁ is the initial length considered.

Fig. 1.

Different stages of X-shaped fractal antenna. The proposed shape is at the stage-4.

Fig. 2.

Geometry of proposed antenna shape.

Different feeding techniques like microstrip feed line, coaxial feed line, proximity coupled, aperture coupled etc. can be used to use this X-shaped fractal as antenna. Special care should be taken in designing the geometry of antenna, while giving power using microstrip line feed so that its strips do not overlap with the feed line or they do not form any closed structure due to overlapping after any iteration.

Therefore, before designing the antenna with this X-shaped fractal, we have derived the condition of no overlapping between strips edges and microstrip feed line. In Fig. 3, the geometry of proposed antenna shape along with one portion zoomed is given, which is required to derive this condition. For simplicity and more clarity, intermediate stages L ₂ and L ₃ are not shown in Fig. 3.

Fig. 3.

The dimensions of the proposed antenna at stage-4. (a) Front view, (b) back view.

It is clearly visible from Fig. 3, that in order to avoid overlapping between strips edges and microstrip feed line, the length AC must be greater than the length DE, i.e.

(2) $$AC\; \gt \; DE$$

From triangle ABC and DBE and with the help of Pythagoras theorem, (2) can be written as –

(3) $$AB\sqrt 2 \; \gt \; BE\sqrt 2. $$

After substituting the value of AB and BE in terms of L ₁, W, and W ₀, (3) becomes –

(4) $$\left( {\displaystyle{{L_1} \over 2} - \displaystyle{W \over 2} - \displaystyle{{W_0} \over {\sqrt 2}}} \right)\sqrt 2 \; \; \; \gt \; \; \left( {\displaystyle{{L_2} \over 2} - \displaystyle{W \over 2} + \displaystyle{{L_3} \over 2} + \displaystyle{{L_4} \over 2} + \displaystyle{W \over 2}} \right)\sqrt 2 \; \;, $$

where, W is the width of strip line.

Now, after substituting the value of L ₂, L ₃, and L ₄ in terms of L ₁ as given in (1), and (4) can be written as

(5) $$1 - m - m^2 - m^3 \gt \displaystyle{W \over {L_1}}\left( {1 + \displaystyle{{W_0} \over W}\sqrt 2 \;} \right)\; \;. $$

The above expression (5) can be generalized for N _th stage to determine the range of scale factor m to avoid the overlapping:

(6) $$1 - \left[ {\mathop \sum \limits_{j = 1}^{N - 1} m^j} \right] \gt \displaystyle{{W_\;} \over {L_1}}\; \left( {1 + \displaystyle{{W_0} \over W}\sqrt 2 \;} \right)\; \;, $$

where, N is the stage number, W ₀ is the width of feed line and L ₁ is the initial length of strip lines at stage-1.

III. ANTENNA DESIGN

The dimensions of proposed antenna based on fractal shape as discussed in Section-II are given in this section. The resonance and impedance can be greatly affected by scale factor m (as it can control the size of the antenna), width of ground plane, number of slots in ground plane and their lengths; hence we can change theses different parameters for getting desired results.

In our proposed antenna, the width of all metal strips is taken identical as 3.0 mm. Microstrip feed line coupling used with the width of feed line is W ₀.The geometry of the X-shape fractal is symmetric to the 50 Ω microstrip fed line. The dimensions of the ground plane are L _g  × W _g . The thickness of substrate (T) is 1.6 mm. Other dimensions of the proposed antenna are listed in Table 1.

Table 1.

Dimensions used in the proposed X-shaped fractal antenna.

As it is known that m must be a real number whose value lies between 0 and 1, the following condition is obtained after substituting the value of W, W ₀, and L ₁ (as given in Table 1) in (6)

$$m \lt 0.5029$$

For the simplicity of design we have taken m = 0.5 in this paper.

IV. RESULTS AND DISCUSSIONS

In this section, the simulation results of proposed antenna are discussed and compared with the experimental results obtained after the fabrication of antenna. In the proposed antenna, patch and ground plane are made up of copper (annealed) of 0.07 mm thickness, electric conductivity of 5.8 × 10⁷ and is fabricated on FR4 substrate, with substrate thickness of 1.6 mm, dielectric constant of 4.4, and loss tangent of 0.02. The parametric results of following cases are discussed:

1. (a) Proposed structure at different stages (Stage-1–4) with constant width of ground plane.

2. (b) Variation in the width of ground plane of proposed antenna (at stage-4).

3. (c) Proposed antenna with DGS (having slots in ground plane).

4. (d) Proposed antenna with reduced feed line length and net reduction in total size

A) X-shaped fractal patch antenna with different stages of patch having constant width of ground plane

To observe the variation in resonant frequency, the reflection coefficient |S ₁₁| is investigated by designing and simulating different stages of X-shaped fractal geometry with the help of simulator, CST MICROWAVE STUDIO 2014. Figure 4 shows the results of reflection coefficient |S ₁₁| for four different stages (Stage-1–4), with width of ground plane W _g as 20 mm as shown in Fig. 2. It can be easily observed from the graph that as the number of stages increases, the number of bands with |S ₁₁ < −10 dB| also increases and as we approach to stage-4, the antenna shows more multiband and wideband behavior. The reason of multiple resonances is the increase in paths and lengths of the antenna geometry. The proposed structure can also be designed beyond the stage-4, but it roughly doubles the antenna size as compared with 4th stage, which is practically difficult to use.

Fig. 4.

Reflection coefficients of the proposed X-fractal antenna in different stages.

B) X-shaped fractal patch antenna with different width of ground plane

Further, to optimize the characteristics of proposed antenna, the width of ground plane is varied and reflection coefficient for various cases has been studied. Figure 5(a) shows the reflection coefficient curve with different width of ground plane (vary from 4 to 28 mm with minimum variation of 4 mm). It is clearly visible from Fig. 5(a), that the proposed antenna structure can be fabricated with any particular ground plane width as it is giving multiband behavior for all the cases.

Fig. 5.

Reflection coefficients of the proposed X-fractal antenna for different width of ground plane, (a) W _g is 4, 8, 12, 16, 20, 24, 28 (b) W _g is 5, 6, 7.

However, the better results are obtaining from W _g  = 4–8 mm. Therefore, we have further investigated the antenna by varying the width of ground plane from 5 to 7 mm  and obtained its reflection coefficient curve as shown in Fig. 5(b). It is visible from Fig. 5(b) that the best results are obtained with W _g  = 6 mm. The antenna structure with W _g  = 6 mm is shown in Fig. 6.

Fig. 6.

(a) Front view, (b) back view of proposed antenna with W _g  = 6 mm.

C) X-shaped fractal patch antenna having slots in ground plane

The results are further optimized using the slots in ground plane. Figure 7 shows the patch and ground plane of width 6 mm having three slots into it. Further, design and simulation of antenna is carried out for different value of slots length in the ground plane for optimized reflection coefficient value. Figure 8 shows the reflection coefficient value for the slot length of 2, 3, and 4 mm in the ground plane. It can be seen that, by cutting the slots of 4 mm in ground plane, the net electrical length increases without changing the input impedance of antenna as shown in smith chart in Fig. 9.

Fig. 7.

(a) Front view, (b) back view of X-fractal antenna with three slots in 6 mm  ground plane.

Fig. 8.

Reflection coefficients of the proposed X-fractal antenna for three slots in the ground plane of different length.

Fig. 9.

Smith chart plot showing input impedance of 49.16 Ω.

Figure 10 shows the photograph of fabricated antenna having X-shaped fractal patch up to 4th iteration, a ground plane of 6 mm width having three slots of 4 mm length. Measurement of results is carried out by E5071C Network Analyzer.

Fig. 10.

Photograph of (a) front and (b) back view of the fabricated antenna.

Figures 11 and 12 shows that the proposed antenna has good impedance match, directivity and radiation efficiency for the desired bands, covering various applications of L, S, and C Frequency bands as shown in Tables 2 and 3.

Fig. 11.

Measured and simulated reflection coefficients of the proposed X-fractal antenna.

Fig. 12.

Simulated 3D radiation patterns of the proposed X-fractal antenna for frequency (a) 2 GHz, (b) 3.5 GHz, (c) 4.9 GHz, (d) 6.5 GHz.

Table 2.

Results of radiation patterns.

Table 3.

Frequency bands and their applications.

D) X-shaped fractal patch antenna with reduced feed line length and net reduction in total size

To utilize wideband feature for practical application and to further reduce total size of antenna, the proposed X-shaped fractal antenna is further optimized by reducing the feed line length with DGS. Different values of feed line lengths are taken and results are carried out. Best optimized results are obtained for antenna having feed line length of 50 mm with net reduction in size of approximately 13% as compared with original antenna, as shown in Figs 13 and 14 (photograph of fabricated antenna with reduced size). Figure 15 shows the smith chart plot of the proposed antenna showing the input impedance value to be 49.11 Ω. Thus, maximum power will be transferred when Coaxial cable of 50 Ω characteristic impedance will be used for feeding the microstrip line.

Fig. 13.

(a) Front view (b) Back view of X-fractal antenna with reduced total size.

Fig. 14.

Photograph of (a) front and (b) back view of the fabricated antenna with reduced size.

Fig. 15.

Smith chart plot of the proposed antenna showing input impedance of 49.11 Ω.

The reflection coefficient value of simulated and fabricated antenna as shown in Fig. 16 shows a good wideband and multiband behavior of presented antenna. Figure 17 shows proposed antenna is also having good directivity and efficiency for the desired frequency bands, which covers various applications in S and C band as shown in Tables 4 and 5.

Fig. 16.

Measured and simulated reflection coefficients of the proposed X-fractal antenna with reduced size.

Fig. 17.

Simulated 3D radiation patterns of the proposed X-fractal antenna with reduced size for frequency (a) 3.6 GHz, (b) 5.5 GHz, (c) 5.95 GHz, (d) 6.5 GHz.

Table 4.

Results of radiation patterns.

Table 5.

Frequency bands and their applications.