**J.K. Xiao and H.F. Huang**

Fractal-shaped microwave passive circuits offer great promise for small circuits with good performance. Planar bandpass filters have been fabricated using equilateral triangular resonators with fractal patch defection to achieve wide passbands and stopband transmission zeros and low passband loss. The fractal defection acts as a perturbation to improve filter performance. Compared with reported dual-mode or multimode filters, newly developed filters offer enhanced performance even though they operate at a single mode. The filters feature compact and simple structures based on the use of a single patch resonator without coupling gaps.

Patch resonators can be used in relatively simple designs to reduce fabrication uncertainties and yield low insertion loss and higher power- handling capabilities compared to line-based resonator filters. Microstrip triangular resonators,^{1-4} especially equilateral types, have been used in microwave circuits, although less so than other resonators such as square and circular forms.

The term "fractal"^{5,6} means broken or fractured; it is derived from a Latin word "fractus." As a branch of classical mathematics, fractal approaches date to the 19th century. Fractal mathematics refers to a total designation of self-similarity figures and structures without characteristic length. Basic fractal geometries can be divided into the following types of Koch fractal structures: Minkowski fractal, Sierpinski and Hilbert fractals, and more; circuits are miniaturized with broadband operation.^{7} Fractal engineering has two basic properties: self-similarity and space filling. Self-similarity means a portion of the fractal geometry looks like that of the entire structure, and space filling means a fractal shape can be filled in a limited region as the order increases without increasing the area. Classical fractal structures generated by mathematic methods have strict self-similarity, and can be called well-regulated fractals; the most commonly applied configurations are irregular fractals with rough self-similarity. There are two types of fractal forms for microwave filters: line-based and patchbased fractal forms. A linebased fractal bandpass filter^{7} has good second-harmonic suppression, electric length increasing without adding circuit size, simple circuit topology, and superior performance compared with filters using defected-ground-structure (DGS) or electromagnetic- bandgap (EBG) structures.

The negative aspects of a fractal filter include its complexity, which can be difficult to simulate and manufacture due to the growing complexity as the fractal order increases. But most line-based fractal filters are modified using traditional microstrip coupled-line structures, and can be difficult to implement with good selectivity and low insertion loss. Fortunately, a patch resonator with fractal-shaped defection can overcome these limitations via a simple configuration and acting as a perturbation that enhances the resonance of a desired mode and suppresses harmonics.

The use of a fractal structure can change the current flow pattern in a filter, with current distributing along the flexural conductor surface rather than on the surface of a simple patch, thus increasing the electrical length. For a highpermittivity substrate, a fractal perturbation may introduce higher-order modes, and suppress undesired harmonics to aid in the implementation of multiband or wideband circuit designs. For a low-permittivity substrate, mode splitting is not easily shown, and higher-order modes are not easy coupled to implement multimode operation since the distance of neighboring resonant modes is too large.

A wideband bandpass filter can be implemented by using fractal perturbation and a low-permittivity substrate because the fractal approach enhances the resonance of the dominant mode. On the other hand, the bandwidth for bandpass filters increases for substrates with decreasing permittivity, when the same filter configurations are compared. To demonstrate the effectiveness of the fractal approach, planar bandpass filters were designed using a patch defected single equilateral triangular resonator. The performance of these fractal filters was an improvement in terms of broadband performance compared to earlier results,^{1,2} achieving more compact sizes without coupling gaps.

Microstrip triangular resonators are applied in microwave circuits, especially the equilateral form, which has exact solution, and resonant frequency can be expressed as8:

f_{r} = {2c/e(e_{r})^{0.5}>}(m^{2} + mn + n^{2}) (1)

where

c = the speed (velocity) of light,

e_{r} = the relative permittivity of the circuit substrate,

ae = an effective value of triangle side length a^{8}, and

m, n = integers that determine the resonant mode.

Using this expression in a commercial electromagnetic (EM) simulation software program, the resonant performances of the dominant mode and a higher-order mode for the equilateral triangular resonator with one fractal defection cell were obtained (* Fig. 1*), using a value of 2.56 for e

_{r}. The resonant frequency of the dominant mode decreases as parameters d and h (data not shown) increase; the contrary is the case for the first higher-order mode. At the same time, the resonant frequencies of the fractal triangular resonator are lower than for a resonator without defection.

The basic principle for designing a microwave patch filter requires determining the required selectivity for a given band and applying different resonant modes. The multimode approach is one of the more efficient methods for implementing a miniature wideband bandpass filter, although a high-permittivity substrate is needed for higher-order modes. With a low-permittivity substrate, it is difficult to introduce dual-mode and multi-mode operation, but parasitic harmonics can be suppressed and the resonance of the dominant mode enhanced by using certain perturbations. Typically, the bandwidth of a bandpass filter will increase as the relative permittivity of the substrate decreases. But a wideband filter can be implemented with a single mode and patch resonator with low-permittivity substrate.

According to the principles detailed earlier, and the calculated results (* Fig. 1*), bandpass filters with the dominant mode operation can be designed by using a fractal equilateral triangular patch resonator, and with a substrate having relative permittivity of 2.56 and a thickness of 0.8 mm. The length of the triangle's side, a, is 15 mm. The input/ output (I/O) feed lines have a characteristic impedance of 50 Ωs and are set at the triangle's hemline (

*).*

**Fig. 2** For filter model 1 with one defection cell, basically a rough loop resonator filter, the triangular defection has rough self-similarity with the triangular resonator; calculated frequency responses are shown in * Fig. 3*. The calculated responses show that a wide passband, a pair of transmission zeros, and low passband insertion loss are obtained, with filter performance greatly improved by the fractal defection for it enhances the resonance of the operation mode. With defection size of b = h = 8 mm and f = 1.5 mm, the filter has a center frequency at 7.1 GHz, and possesses a relative 3-dB bandwidth of 38.6 percent, and passband insertion loss of no more than 0.4 dB

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* Figure 3* also shows that parameter f (the distance of the I/O feed lines) has some effect on filter bandwidth, return loss, and the attenuation of transmission zeros, for there is a coupling between the I/O lines. The degree of coupling (which may be denoted with a capacitor) can be adjusted, so there is more control over the parameter than for a filter feed method based on triangle bevel edge feeding.

Filter model 2 has a loop fractal defection with defection size reduced compared with filter model 1, but with electric length increased. The simulated frequency response for this design has a wide passband with relative bandwidth of 38 percent and passband insertion loss of no more than 0.3 dB; the design provides overall better performance than model 1 due to the added electric length.

If the number and size of the defection cells are increased, the space-filling property of the fractal approach can be enhanced as well as the electric length.* Figure 2(c)* shows the proposed filter model with three identical defection cells, with the frequency responses for the three illustrated in

*. The performance of filter upper stopband improved for the three-cell designs compared for that of one defection cell, and the three-cell filters have relative bandwidth of about 30 percent, low passband insertion loss of 0.4 dB or less at a center frequency of about 6 GHz, and multi transmission zeros. To verify the designs, models 1 and 2 were fabricated and measured, with results shown in*

**Fig. 4***and*

**Figs. 5***, respectively. Measurements agreed closely with simulations.*

**6** ACKNOWLEDGMENT

This project was supported by the Specialized Research Fund for China Doctoral Program of Higher Education (grant No. 200805611077).

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