**7. Fabricated model and the measured results**

*Fractal Analysis*

plays a role in causing the resonance.

**6. Comparison with other fractal-based filter models**

other fractal geometries with high space-filling properties.

*Moore fractal-based DGS BPFs together with performance responses.*

*Peano fractal-based DGS BPFs together with performance responses.*

What is more, to offering an additional physical clarification about the electromagnetic aspects of the modeled band-pass filter, the current distributions on its surface have been simulated at different frequencies in the resonant band and outside it. **Figure 15(a)**–**(c)** illustrate the surface current distributions at 1.30, 1.55, and 1.70 GHz which represent the frequencies in the lower stopband, in the passband, and in the upper stopband, respectively. **Figure 15** exhibits the current distributions on the surfaces of the ground plane of the modeled BPF structure. **Figure 15(a)** and **(c)** indicates that there is no coupling taking place between the resonators in the lower stopband and the upper stopband. On the other hand, the large current densities, exposed in **Figure 15(b)**, represent an indication of the strong coupling which results in the conclusive resonance. It is apparent that the majority of the resonator length

In this chapter, the modified Minkowski fractal geometry has been adopted to design the proposed BPF filter. The modified Minkowski fractal geometry is with better space-filling property to achieve more miniaturization as compared with the conventional Minkowski fractal geometry. However, an attempt has been carried out to compare the performance of the proposed filter with those modeled using

For this purpose, Peano and Moore fractal geometries have been adopted to design two BPFs based on the presented design idea. In the modeling of the proposed BPF filters, the same substrate and the same resonator dimensions are used. **Figures 16** and **17** illustrate the filter structures together with their performance

**42**

**Figure 17.**

**Figure 16.**

A prototype of the fractal-based defected ground structure band-pass filter has been manufactured. The fabricated prototype uses an identical substrate with a relative permittivity of 2.65 and thickness of 1.0 mm. **Figure 18** shows photos of the manufactured filter. The measured and simulated scattering coefficient responses, *S*11 and *S*21, are depicted in **Figures 19** and **20**, respectively.

The simulated and measured results of the modeled and the fabricated bandpass filters well agree with each other. Some deviation, between the measured and the simulated results, is noticed. The shift of the lower edge of the passband response of the *S*21 responses is slight, while that of the upper edge is hardly visible. Furthermore, the measured results reveal attenuation in the passband region.

The production technique might cause dimensional tolerance which, in turns, leads to the differences observed between the measured and the simulated results. Employing more advanced manufacturing methods, besides the selection of a substrate having a more stable parameter, will result in a closer agreement.

**Figure 18.** *Photos of the fabricated prototype (a) the top and (b) the bottom views.*

**Figure 19.** *The simulated and measured S11 responses of the fabricated filter prototype.*

**Figure 20.**

*The simulated and measured S21 responses of the fabricated filter prototype.*

The resonant behaviors of the modeled band-pass filters suggested in this work have to be compared with those recently reported in the literature. **Table 2** summarizes a comparison of the performances of the presented filters which are based on the zero, first, and second iteration DGS band-pass filters with those recently reported in the literature [27, 28]. As the table implies, the comparison is carried out


**45**

*Fractal Geometry: An Attractive Choice for Miniaturized Planar Microwave Filter Design*

concerning the occupied areas and the filter selectivity which in turns is expressed by the roll-off rates at the lower and the upper edges. In spite of the extra miniaturization of the second iteration, fractal-based DGS band-pass filter is minor, but this BPF possesses the most excellent selectivity, among the others, regarding the lower and the upper edges roll-off rates. However, the realized size by each BPF has been calculated in terms of the guided wavelength, *λ*g, computed at the lower resonant frequency. Even though the DGS band-pass filter reported in [28] is approximately equivalent in the occupied area with that suggested in this work, it suffers from poor selectivity. It has poor upper edge roll-off rate and low selectivity in the lower

The defected ground structure resonator based on the Minkowski fractal variant reported in this chapter has confirmed its capability to produce reduced size microstrip band-pass filters. Besides the acceptable resonant responses of the suggested BPFs, the adoption of the Minkowski fractal geometry to the defected ground structure resonator bring about BPF designs with considerable miniaturization with reference to those recently published in the literature. As expected, the results showed that more filter size miniaturization could be obtained when employing higher fractal orders. In the real practice, this might not be the situation; several restrictions are coming across the practical implementation of a filter prototype, especially for the higher iteration levels. Also, the results revealed that the proposed BPF performances are characterized by a low loss in the passband and high rejection in the stopband with considerable reduction of higher harmonics. A significant finding is that the final BPF performance possesses a high selectivity with steep roll-off rates at both the lower and the upper edges of the passband. A comparison of the performance of the DGS BPF based on the modified Minkowski fractal geometry with other filters based on Peano and Hilbert fractal geometries revealed that the proposed BPF has acceptable resonant responses with a significant lessening of higher harmonics. Measured results of a fabricated prototype well agree with those evaluated by the EM simulator. The presented BPF resonant characteristics, besides the considerable size miniaturization, will make it an appropriate candidate for the application in a broad diversity of the modern wire-

The authors declare there are no conflicts of interest regarding the publication of

*DOI: http://dx.doi.org/10.5772/intechopen.81353*

edge roll-off rate.

**8. Conclusions**

less communication services.

**Conflict of interest**

this book chapter.

#### **Table 2.**

*Comparison of the presented band-pass filters with those published in the literature.*

*Fractal Geometry: An Attractive Choice for Miniaturized Planar Microwave Filter Design DOI: http://dx.doi.org/10.5772/intechopen.81353*

concerning the occupied areas and the filter selectivity which in turns is expressed by the roll-off rates at the lower and the upper edges. In spite of the extra miniaturization of the second iteration, fractal-based DGS band-pass filter is minor, but this BPF possesses the most excellent selectivity, among the others, regarding the lower and the upper edges roll-off rates. However, the realized size by each BPF has been calculated in terms of the guided wavelength, *λ*g, computed at the lower resonant frequency. Even though the DGS band-pass filter reported in [28] is approximately equivalent in the occupied area with that suggested in this work, it suffers from poor selectivity. It has poor upper edge roll-off rate and low selectivity in the lower edge roll-off rate.

## **8. Conclusions**

*Fractal Analysis*

**44**

**Table 2.**

**Figure 20.**

**Figure 19.**

The resonant behaviors of the modeled band-pass filters suggested in this work have to be compared with those recently reported in the literature. **Table 2** summarizes a comparison of the performances of the presented filters which are based on the zero, first, and second iteration DGS band-pass filters with those recently reported in the literature [27, 28]. As the table implies, the comparison is carried out

Zero iteration DGS 0.45 × 0.23 78.51 47.10 First iteration DGS 0.30 × 0.15 132.74 94.81

DGS BPF [27] 0.50 × 0.25 72.54 52.86 DGS BPF [28] 0.22 × 0.17 41.11 12.75

*Comparison of the presented band-pass filters with those published in the literature.*

**Roll-off rate (lower edge) (dB/GHz)**

0.29 × 0.14 197.70 180.04

**Roll-off rate (upper edge) (dB/GHz)**

*The simulated and measured S21 responses of the fabricated filter prototype.*

*The simulated and measured S11 responses of the fabricated filter prototype.*

**[λg] 2**

**Filter type Filter size** 

Second iteration

DGS

The defected ground structure resonator based on the Minkowski fractal variant reported in this chapter has confirmed its capability to produce reduced size microstrip band-pass filters. Besides the acceptable resonant responses of the suggested BPFs, the adoption of the Minkowski fractal geometry to the defected ground structure resonator bring about BPF designs with considerable miniaturization with reference to those recently published in the literature. As expected, the results showed that more filter size miniaturization could be obtained when employing higher fractal orders. In the real practice, this might not be the situation; several restrictions are coming across the practical implementation of a filter prototype, especially for the higher iteration levels. Also, the results revealed that the proposed BPF performances are characterized by a low loss in the passband and high rejection in the stopband with considerable reduction of higher harmonics. A significant finding is that the final BPF performance possesses a high selectivity with steep roll-off rates at both the lower and the upper edges of the passband. A comparison of the performance of the DGS BPF based on the modified Minkowski fractal geometry with other filters based on Peano and Hilbert fractal geometries revealed that the proposed BPF has acceptable resonant responses with a significant lessening of higher harmonics. Measured results of a fabricated prototype well agree with those evaluated by the EM simulator. The presented BPF resonant characteristics, besides the considerable size miniaturization, will make it an appropriate candidate for the application in a broad diversity of the modern wireless communication services.
