**3. Fractal and polar transformations**

### **3.1 Types and applications**

From a mathematical point of view, a fractal refers to a set in Euclidean space with specific properties, such as self-similarity or self-affinity, simple and recursive definition, fractal dimension, irregular shape, and natural appearance [27]. Fractal geometry is the study of sets with these properties, which are too irregular to be described by calculus or traditional Euclidian geometry language [27, 28].

Fractals are resorted to conventional classes, such as geometrical fractals, algebraic fractals, and stochastic fractals [29]. Two common methods used to generate mathematical fractals are iterated function systems (IFS) and Lindenmayer systems [27–30].

IFS method used to generate a 2-D fractal consisting of a collection of affine transformations with probability given by (1). Affine transformations are most commonly used in IFS. The coefficients of a two-dimensional affine transformation represent the IFS code for scaling, rotations, and translations. An affine transformation of a point to the point is given in (2).

$$\begin{array}{ll}\text{for a point } \mathbf{r} \text{ the point is given in } (\downarrow). \\\\ \begin{cases} T\_1 \colon & \{a\_1, b\_1, c\_1, d\_1, e\_1, f\_1, P\_1\} \\ T\_2 \colon & \{a\_2, b\_2, c\_2, d\_2, d\_2, e\_2, f\_2, P\_2\} \\ \vdots & \vdots \\ T\_m \colon & \{a\_m, b\_m, c\_m, d\_m, e\_m, f\_m, P\_m, P\_m\} \end{cases} \\\\ \begin{cases} \mathcal{X}\_{n+1} = a \mathcal{X}\_n + b \mathcal{y}\_n + e \\\ g\_{n+1} = c \mathcal{x}\_n + d \mathcal{y}\_n + f \end{cases} \\ \tag{2}$$

IFS algorithm consists of four steps: (i) start with an arbitrary point in the plane p0 = (x0, y0); (ii) pick a random transformation, Tm, according to the probabilities, Pm; (iii) transform the point p1 = Tm (p0) and plot it; and (iv) go to step 2. IFS algorithm is continued ad infinitum (for ideal fractal) or until a given number of fractal iterations is reached (for pre-fractals).

Lindenmayer system (or L-system) was initially conceived to model growth phenomena in biological organisms [31]. An L-system grammar handles an initial string of symbols (axiom) and includes a set of production rules that may be applied to the symbols (letters of the L-system alphabet) to generate new strings. A graphic interpretation of strings, based on turtle geometry, is described in [29, 32]. A state of the turtle is defined as a triplet (xk, yk, φk) where coordinates (xk, yk, φk) and angle φk represent the turtle's position and direction, respectively, (3).

##  $(\cdots, \searrow, \cdots)$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .  $\omega$ .

**157**

up to *k* = 32 petals.

**Figure 8.**

**Figure 9.**

*Island.*

*v*

*Fractal and Polar Microstrip Antennas and Arrays for Wireless Communications*

that is, for each real value, *t* is associated with a vector in ℜ<sup>2</sup>

*Esthetic polar transformation for k varying up to k = 32 petals in (5).*

*<sup>v</sup>*

<sup>→</sup>(*t*) <sup>=</sup> (<sup>1</sup> <sup>+</sup> cos(*t*) \_\_\_\_\_\_ <sup>2</sup> ) <sup>⋅</sup> (cos(

**4.1 Koch fractal microstrip antenna**

**4. Fractal and polar-shaped microstrip antenna**

esthetic polar transformation defined by (5) is presented in **Figure 9** for k varying

*IFS and L-system pre-fractals: (a) Koch curve; (b) modified Barnsley fern; (c) Koch Island; (d) Minkowski* 

<sup>→</sup>(*t*):*I* → ℜ*<sup>n</sup> t* → *v* <sup>→</sup>(*t*)

The design of pre-fractals patch antennas has been a subject of great interest to designers and researchers in the field of antennas. Previously published

*<sup>k</sup>* ), sin(

\_\_\_\_\_\_\_\_\_ 2*t* − *sen*(2*t*)

\_\_\_\_\_\_\_\_\_ 2*t* − *sen*(2*t*)

, (4). An example of an

*<sup>k</sup>* )), <sup>0</sup> <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>k</sup>* (5)

(4)

*DOI: http://dx.doi.org/10.5772/intechopen.83401*

The simplest class of L-systems is termed deterministic and context-free or DOL-systems [29, 32]. DOL-system is defined as a triple *H = (V, ω*, Π*)*, where *V* is the L-system alphabet, *ω* is the axiom word, and Π is a finite set of productions. Formal definitions of D0L-systems and their operation can be found in [29, 32]. Given the initial state of turtle (*x0*, *y0*, *φ0*), step size *d*, and the angle increment Δ*φ*, the turtle can respond to the commands in L-system strings *<sup>v</sup>* <sup>∈</sup> *<sup>V</sup>* and represented by the following symbols [29, 32]:

F → Move forward a step of length *d*, and change state of the turtle according to (3). A line segment between points (*xk*, *yk*) and (*xk + 1*, *yk + 1*) is drawn.

f → Move forward a step *d* without drawing a line. The turtle state changes as above.

+ → Turn right by angle Δ*φ*. The next state of the turtle is given by (*xk*, *yk,* Δk + *φk*).


transformation is defined in this chapter through a vector function *v* <sup>→</sup>(*t*) <sup>=</sup> (*x*(*t*),*y*(*t*)), *<sup>t</sup>* <sup>≥</sup> 0,

*Fractal and Polar Microstrip Antennas and Arrays for Wireless Communications DOI: http://dx.doi.org/10.5772/intechopen.83401*

*Wireless Mesh Networks - Security, Architectures and Protocols*

tion of a point to the point is given in (2).

{

tion, respectively, (3).

{

the following symbols [29, 32]:

⎧ ⎪ ⎨ ⎪ ⎩

fractal iterations is reached (for pre-fractals).

[27–30].

Fractals are resorted to conventional classes, such as geometrical fractals, algebraic fractals, and stochastic fractals [29]. Two common methods used to generate mathematical fractals are iterated function systems (IFS) and Lindenmayer systems

IFS method used to generate a 2-D fractal consisting of a collection of affine transformations with probability given by (1). Affine transformations are most commonly used in IFS. The coefficients of a two-dimensional affine transformation represent the IFS code for scaling, rotations, and translations. An affine transforma-

> *T*<sup>1</sup> : (*a*1, *b*1, *c*1, *d*1, *e*1, *f*1, *P*1) *<sup>T</sup>*<sup>2</sup> : (*a*2, *<sup>b</sup>*2, *<sup>c</sup>*2, *<sup>d</sup>*2, *<sup>e</sup>*2, *<sup>f</sup>*2, *<sup>P</sup>*2) <sup>⋮</sup> <sup>⋮</sup>

*Tm* : (*am*, *bm*, *cm*, *dm*, *em*, *fm*, *Pm*)

*xn*+1 = *axn* + *byn* + *e*

IFS algorithm consists of four steps: (i) start with an arbitrary point in the plane p0 = (x0, y0); (ii) pick a random transformation, Tm, according to the probabilities, Pm; (iii) transform the point p1 = Tm (p0) and plot it; and (iv) go to step 2. IFS algorithm is continued ad infinitum (for ideal fractal) or until a given number of

Lindenmayer system (or L-system) was initially conceived to model growth phenomena in biological organisms [31]. An L-system grammar handles an initial string of symbols (axiom) and includes a set of production rules that may be applied to the symbols (letters of the L-system alphabet) to generate new strings. A graphic interpretation of strings, based on turtle geometry, is described in

[29, 32]. A state of the turtle is defined as a triplet (xk, yk, φk) where

coordinates (xk, yk, φk) and angle φk represent the turtle's position and direc-

*xk*+1 = *xk* + *d* cos(φ*k*) *yk*+1 <sup>=</sup> *xk* <sup>+</sup> *<sup>d</sup>* sin(φ*k*)

F → Move forward a step of length *d*, and change state of the turtle according to


f → Move forward a step *d* without drawing a line. The turtle state changes as

+ → Turn right by angle Δ*φ*. The next state of the turtle is given by (*xk*, *yk,*

(3). A line segment between points (*xk*, *yk*) and (*xk + 1*, *yk + 1*) is drawn.

transformation is defined in this chapter through a vector function *v*

The simplest class of L-systems is termed deterministic and context-free or DOL-systems [29, 32]. DOL-system is defined as a triple *H = (V, ω*, Π*)*, where *V* is the L-system alphabet, *ω* is the axiom word, and Π is a finite set of productions. Formal definitions of D0L-systems and their operation can be found in [29, 32]. Given the initial state of turtle (*x0*, *y0*, *φ0*), step size *d*, and the angle increment Δ*φ*, the turtle can respond to the commands in L-system strings *<sup>v</sup>* <sup>∈</sup> *<sup>V</sup>* and represented by

*yn*+1 <sup>=</sup> *cxn* <sup>+</sup> *dyn* <sup>+</sup> *<sup>f</sup>* (2)

(1)

(3)

<sup>→</sup>(*t*) <sup>=</sup> (*x*(*t*),*y*(*t*)), *<sup>t</sup>* <sup>≥</sup> 0,

**156**

above.

Δk + *φk*).

*IFS and L-system pre-fractals: (a) Koch curve; (b) modified Barnsley fern; (c) Koch Island; (d) Minkowski Island.*

**Figure 9.** *Esthetic polar transformation for k varying up to k = 32 petals in (5).*

that is, for each real value, *t* is associated with a vector in ℜ<sup>2</sup> , (4). An example of an esthetic polar transformation defined by (5) is presented in **Figure 9** for k varying up to *k* = 32 petals.

$$\begin{array}{c} \vec{v}(t) \colon I \to \mathfrak{R}^n \\ t \to \vec{v}(t) \end{array} \tag{4}$$

$$\vec{\upsilon}(t) = \left(\mathbb{1} + \frac{\cos(t)}{2}\right) \cdot \left(\cos\left(\frac{2t - \operatorname{sen}(2t)}{k}\right), \sin\left(\frac{2t - \operatorname{sen}(2t)}{k}\right)\right), \ 0 \le t \le k\pi \tag{5}$$

### **4. Fractal and polar-shaped microstrip antenna**

### **4.1 Koch fractal microstrip antenna**

The design of pre-fractals patch antennas has been a subject of great interest to designers and researchers in the field of antennas. Previously published

**Figure 10.**

*Pre-fractals patch antennas: (a) image layouts; (b) parametric analysis; (c) neuromodeling; (d) image comparing overall sizes of the built patch antennas—Rectangular and pre-fractals.*

works by the authors have contributed to this research area, showing the miniaturization of inset-fed patch antennas with the use of Koch and Minkowski pre-fractals [13, 19, 33], **Figure 10(a)**. Frequency compression factors of 26.1, 39, and 42% were observed for level 2 pre-fractals: triangular Koch, rectangular Koch, and Minkowski, respectively [13, 19, 33]. Pre-fractal patch antennas are defined with two fractal parameters: iteration number (level) and scaling factor. They possess a large design region of interest, **Figure 10(b)**; are easy to model using neural network, **Figure 10(c)**, [19]; and their shapes and multiband behavior facilitate frequency reconfiguration [5]. The unique properties of geometric fractals are useful to synthesis of more compact patch antennas, **Figure 10(d)**, [13, 19, 33, 34].

### **4.2 Wearable teragon antennas**

The use of wearable antennas is necessary that have some characteristics as: easy interaction with the body, low visual impact, preferably low cost, and flexible structure [19]; for this reason, the materials used in the manufacture of the wearable antennas must follow some requirements: easy interaction with the body, flexible structure, reduced visual impact, and preferably low cost [19].

**159**

**Figure 12.**

*Results of teragon antenna: (a) measured, (b) gain in dBi.*

**Figure 11.**

*Fractal and Polar Microstrip Antennas and Arrays for Wireless Communications*

results for reflection coefficient and gain are shown in **Figure 12**.

Teragon was a term coined by Mandelbrot that literally means, "monster curve" [28]. The proposed wearable teragon patch antennas are based on a square patch antenna with displaced microstrip line feed. Square patch antenna dimensions are calculated according to [13, 28, 34]. Pre-fractal teragons were developed with a scale factor of R = 6 and number of copies, n = 18. **Figure 11(a)** shows dimensions and shapes of the teragons. Images of built antenna prototypes with polyamide flexible dielectric substrate are shown in **Figure 11(b)**. Obtained simulated and measured

**Figure 12a** shows the comparison between simulated and measured reflection coefficient of the wearable flexible antennas. The increase of the patch perimeter by the use of teragon shapes provides a reduction of the resonant frequencies. The main highlight is for teragon 1, with reduction of approximately 1 GHz, when

The gain (dB) simulated in resonant frequencies of the wearable path antennas is shown in **Figure 12b**. As noted, the gain is reduced when fractal level increase. The initial square patch antenna presented higher gain, with maximum gain in endfire direction of 6.13 dBi, and the teragon 1 showed the maximum gain of 4.26 dBi

*Pre-fractal teragon patch antenna design steps (a) Matlab dxf images, (b) layouts, (c) prototypes.*

*DOI: http://dx.doi.org/10.5772/intechopen.83401*

compared to the initial square patch antenna.

(**Figure 12b**).

*Fractal and Polar Microstrip Antennas and Arrays for Wireless Communications DOI: http://dx.doi.org/10.5772/intechopen.83401*

Teragon was a term coined by Mandelbrot that literally means, "monster curve" [28]. The proposed wearable teragon patch antennas are based on a square patch antenna with displaced microstrip line feed. Square patch antenna dimensions are calculated according to [13, 28, 34]. Pre-fractal teragons were developed with a scale factor of R = 6 and number of copies, n = 18. **Figure 11(a)** shows dimensions and shapes of the teragons. Images of built antenna prototypes with polyamide flexible dielectric substrate are shown in **Figure 11(b)**. Obtained simulated and measured results for reflection coefficient and gain are shown in **Figure 12**.

**Figure 12a** shows the comparison between simulated and measured reflection coefficient of the wearable flexible antennas. The increase of the patch perimeter by the use of teragon shapes provides a reduction of the resonant frequencies. The main highlight is for teragon 1, with reduction of approximately 1 GHz, when compared to the initial square patch antenna.

The gain (dB) simulated in resonant frequencies of the wearable path antennas is shown in **Figure 12b**. As noted, the gain is reduced when fractal level increase. The initial square patch antenna presented higher gain, with maximum gain in endfire direction of 6.13 dBi, and the teragon 1 showed the maximum gain of 4.26 dBi (**Figure 12b**).

**Figure 12.**

*Results of teragon antenna: (a) measured, (b) gain in dBi.*

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*Pre-fractals patch antennas: (a) image layouts; (b) parametric analysis; (c) neuromodeling; (d) image* 

works by the authors have contributed to this research area, showing the miniaturization of inset-fed patch antennas with the use of Koch and Minkowski pre-fractals [13, 19, 33], **Figure 10(a)**. Frequency compression factors of 26.1, 39, and 42% were observed for level 2 pre-fractals: triangular Koch, rectangular Koch, and Minkowski, respectively [13, 19, 33]. Pre-fractal patch antennas are defined with two fractal parameters: iteration number (level) and scaling factor. They possess a large design region of interest, **Figure 10(b)**; are easy to model using neural network, **Figure 10(c)**, [19]; and their shapes and multiband behavior facilitate frequency reconfiguration [5]. The unique properties of geometric fractals are useful to synthesis of more compact patch antennas,

The use of wearable antennas is necessary that have some characteristics as: easy interaction with the body, low visual impact, preferably low cost, and flexible structure [19]; for this reason, the materials used in the manufacture of the wearable antennas must follow some requirements: easy interaction with the body, flexible

*comparing overall sizes of the built patch antennas—Rectangular and pre-fractals.*

structure, reduced visual impact, and preferably low cost [19].

**158**

**Figure 10.**

**Figure 10(d)**, [13, 19, 33, 34].

**4.2 Wearable teragon antennas**

**Figure 13.** *Wearable textile antennas: (a) L-systems, (b) polar transformer.*

Several shapes were used in development of the microstrip antennas; the polar transformer is the possibility in this case. **Figure 13** shows the wearable textile antennas: patch generated by L-systems (**Figure 13(a)**) and printed monopole generated by polar transformer, **Figure 13(b)**. Printed monopole antennas (PMA) with polar shape can be observed in several works, operating mainly in the ultrawideband (UWB), but with projects for 2G, 3G, and 4G technology and X band [35–39]. The altering frequency provided by polar shapes was observed in [38, 40], similar to the observed pre-fractal geometry applied to the PMA [41].

### **4.3 Polar microstrip antenna**

**Figure 14** shows frequency resonance of polar microstrip antenna for kinteractions (k = 1, 8 12, 16, 24, 32, 40, 48, 56, 64) and the comparison of measured

### **Figure 14.**

*Interactions of polar microstrip patch antenna: a) comparison of frequency resonance simulated; b) comparison of measured antennas.*

**161**

**Figure 16.**

*Fractal and Polar Microstrip Antennas and Arrays for Wireless Communications*


**Figure 14** shows the |S11| parameters measured by the polar antennas to k = 2, 8, 16, 24. We noted than the increase of the patch perimeter by the use of polar interaction provides a reduction of the resonant frequencies, similar to the fractal comportment. The greater difference can be observed in k = 2 and k = 8, of 3.4 GHz,

**Figure 15** shows the use of polar transformer in the development of the array patch antenna with 4 petals, k = 8 interactions. The polar array presented good response, with simulated and measured results closed, had loss return less than −45 dB and bandwidth of 101 MHz, and covered the WLAN band in 2.4 GHz.

The other shape used was the leaf clover, generated by (8). **Figure 16** shows the comparison of |S11| parameter measured and simulated with leaf clover with four and

wave transformer, with dimensions calculated according [7, 9].

*Polar leaf clover patch antennas: (a) four petals, (b) array of two elements with four petals.*

and all structures with dual-frequency resonances.

*DOI: http://dx.doi.org/10.5772/intechopen.83401*

*Polar microstrip patch antenna array with two elements.*

**Figure 15.**

*Fractal and Polar Microstrip Antennas and Arrays for Wireless Communications DOI: http://dx.doi.org/10.5772/intechopen.83401*

**Figure 15.** *Polar microstrip patch antenna array with two elements.*

**Figure 16.**

*Wireless Mesh Networks - Security, Architectures and Protocols*

*Wearable textile antennas: (a) L-systems, (b) polar transformer.*

Several shapes were used in development of the microstrip antennas; the polar transformer is the possibility in this case. **Figure 13** shows the wearable textile antennas: patch generated by L-systems (**Figure 13(a)**) and printed monopole generated by polar transformer, **Figure 13(b)**. Printed monopole antennas (PMA) with polar shape can be observed in several works, operating mainly in the ultrawideband (UWB), but with projects for 2G, 3G, and 4G technology and X band [35–39]. The altering frequency provided by polar shapes was observed in [38, 40],

similar to the observed pre-fractal geometry applied to the PMA [41].

**Figure 14** shows frequency resonance of polar microstrip antenna for kinteractions (k = 1, 8 12, 16, 24, 32, 40, 48, 56, 64) and the comparison of measured

*Interactions of polar microstrip patch antenna: a) comparison of frequency resonance simulated; b)* 

**4.3 Polar microstrip antenna**

**Figure 13.**

**160**

**Figure 14.**

*comparison of measured antennas.*

*Polar leaf clover patch antennas: (a) four petals, (b) array of two elements with four petals.*


**Figure 14** shows the |S11| parameters measured by the polar antennas to k = 2, 8, 16, 24. We noted than the increase of the patch perimeter by the use of polar interaction provides a reduction of the resonant frequencies, similar to the fractal comportment. The greater difference can be observed in k = 2 and k = 8, of 3.4 GHz, and all structures with dual-frequency resonances.

**Figure 15** shows the use of polar transformer in the development of the array patch antenna with 4 petals, k = 8 interactions. The polar array presented good response, with simulated and measured results closed, had loss return less than −45 dB and bandwidth of 101 MHz, and covered the WLAN band in 2.4 GHz.

The other shape used was the leaf clover, generated by (8). **Figure 16** shows the comparison of |S11| parameter measured and simulated with leaf clover with four and six petals for one and two patch elements and prototype images; **Table 3** presents the dimensions used. The polar antenna with six petals presented great bandwidth (52 MHz) than the polar antenna with four petals (42 MHz) and best loss return (−26.3 dB), **Figure 13(b)**.

$$r = 4.4 \text{ - min} \left( abs \left( \tan \left( 2t + \pi/1 \right) \right) / 10, 3 \right) \tag{6}$$

From the leaf clover antennas, polar array patch antennas with two and four elements have been developed. **Figure 16** shows polar array patch antennas with the shape of clover of four and six petals, with two and four elements, operating in WLAN range. The antennas presented measured bandwidth of 81 MHz and half power beamwidth (HPBW) of 55°, the inclination of radiation pattern indicating the great element used in the patch array (**Figure 17**).

**Figure 18** shows the Koch fractal patch antenna array with two elements of the square geometry until Koch level 2. The applications of Koch fractal in the array


### **Table 3.**

*Dimensions of leaf clover antenna (mm).*

**163**

patch element.

**Figure 19.**

**Figure 18.**

*Koch array patch antenna with two elements.*

**5. Conclusions**

*Fractal and Polar Microstrip Antennas and Arrays for Wireless Communications*

structure provide great bandwidth (113 MHz) and maximum gain in end-fire direction of 7.93 dBi (**Figure 19**), with variation of radiation pattern, indicating lager

*Gain comparison of the Koch array patch antenna with two elements.*

In this chapter, we have described some trends for the computer-aided design of microstrip antennas (patches and printed monopoles) for wireless sensors network applications. With the use of such CAD tools, innovative designs of antennas and arrays with pre-fractals and polar motifs were approached and

*DOI: http://dx.doi.org/10.5772/intechopen.83401*

**Figure 17.** *Polar leaf clover patch antennas: (a) six petals, (b) array of two elements with six petals.*

*Fractal and Polar Microstrip Antennas and Arrays for Wireless Communications DOI: http://dx.doi.org/10.5772/intechopen.83401*

**Figure 18.** *Koch array patch antenna with two elements.*

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the great element used in the patch array (**Figure 17**).

*Polar leaf clover patch antennas: (a) six petals, (b) array of two elements with six petals.*

(−26.3 dB), **Figure 13(b)**.

six petals for one and two patch elements and prototype images; **Table 3** presents the dimensions used. The polar antenna with six petals presented great bandwidth (52 MHz) than the polar antenna with four petals (42 MHz) and best loss return

*r* = 4.4 − min(*abs*(tan(2*t* + *π*/1))/10, 3) (6)

From the leaf clover antennas, polar array patch antennas with two and four elements have been developed. **Figure 16** shows polar array patch antennas with the shape of clover of four and six petals, with two and four elements, operating in WLAN range. The antennas presented measured bandwidth of 81 MHz and half power beamwidth (HPBW) of 55°, the inclination of radiation pattern indicating

**Figure 18** shows the Koch fractal patch antenna array with two elements of the square geometry until Koch level 2. The applications of Koch fractal in the array

**162**

**Figure 17.**

**Table 3.**

*Dimensions of leaf clover antenna (mm).*

**Figure 19.** *Gain comparison of the Koch array patch antenna with two elements.*

structure provide great bandwidth (113 MHz) and maximum gain in end-fire direction of 7.93 dBi (**Figure 19**), with variation of radiation pattern, indicating lager patch element.
