II. THE FILTER STRUCTURE

The structure of the proposed filter consists of two parts; the first part is the conventional microstrip dual-mode BPF configuration, which constitutes the upper side of the whole structure, whereas the second part represents the proposed CSRR as a DGS in the ground plane. In this paper, the Minkowski-like pre-fractal geometry has been applied to both parts. Figure 1 demonstrates the generation process of the Minkowski-like pre-fractal geometry as applied to a squared shape ring.

Fig. 1.

The generation process of the Minkowski-like pre-fractal structure: (a) the generator, (b) the square ring, (c) the first iteration, and (d) the second iteration [23].

For simplicity, the same fractal geometry has been applied to both the dual-mode resonator filter and CSRRs DGS. It has been shown that shape modification of the structure depicted in Figs. 1(c) and 1(d) are ways to increase the surface current path length compared with that of the conventional square ring resonator; resulting in a reduced resonant frequency or a reduced resonator size, if the design frequency is to be maintained [Reference Ali23]. For the nth iteration, the Minkowski-like pre-fractal structures depicted in Figs. 1(c) and 1(d) have been found to have the perimeters given by [Reference Ali23, Reference Lee, Soh, Hashim, Vandenbosch, Volski, Adam, Mirza and Aziz32, Reference Alqaisy, Ali, Chakrabarty and Hock33]

(1)$${P_n}=\left({1+2\displaystyle{{{w_2}} \over {{L_ 0}}}} \right){P_{n - 1}}\comma \;$$

where P _n is the perimeter of the nth iteration pre-fractal structure, and w ₂ and L ₀ are as depicted in Fig. 1. Equation (1) and Fig. 1 imply that, at certain iteration level, a wide variety of structures with different perimeters can be obtained by varying w ₁, w ₂, or both.

It is expected then that further miniaturization can be achieved when applying fractal geometries to the conventional square CSRR. This means that higher iterations will result in further miniaturization possibility owing to its extra space filling property. The increase in length decreases the required volume occupied for the pre-fractal bandpass filter at resonance. Theoretically, as n goes to infinity the perimeter goes to infinity. The ability of the resulting structure to increase its perimeter in the successive iterations was found very triggering for examining its size reduction capability as a microstrip bandpass filter. It should be reminded again that the same fractal has been applied to the dual-mode ring resonator (on the top of the filter structure) and the CSRR in the ground plane (on the bottom of the filter structure). However, this does not prevent the possibility to use other types of space filling fractal curves for the same task.

III. THE FILTER DESIGN

It is worth to mention here that, a dual-mode fractal-based ring resonator, as shown in Fig. 2(a), has been adopted by one of the authors to produce a miniaturized dual-mode microstrip BPF [Reference Ali23], where the Minkowski-like fractal geometry has been applied for this purpose. In this paper, the same dual-mode filter structure is recalled again. What is new here is that the filter ground plane is perforated with fractal-based CSRRs shown in Figs. 1(c) and 1(d) in an attempt to produce a miniaturized dual-band dual-mode BPF. Figure 2 demonstrates the details of the proposed filter structure. The top view of this structure, Fig. 2(a), represents the dual-mode BPF with the associated input/output feed lines. The ring resonator of this filter has the structure corresponding to the second iteration Minkowski pre-fractal geometry. The bottom views, Figs. 2(b) and 2(c), show the fractal-based CSRR structures corresponding to the first and second iterations Minkowski pre-fractal geometry that have to be embedded in the proposed filter ground plane.

Fig. 2.

The proposed microstrip dual-band dual-mode BPF structure: (a) the dual-mode fractal-based ring structure (top view), (b) and (c) CSRRs with Minkowski fractal-shaped inner ring of the first and second iterations, respectively (bottom views).

The first step, the filter structure of Fig. 2(a) has been designed for the ISM band applications at 2.45 GHz with its ground plane not defected. The filter structure has been etched using a substrate with a relative permittivity of 2.2 and thickness of 0.787 mm. The input/output ports have characteristic impedances of 50 Ω. This corresponds to a transmission line width of about 1.12 mm. At resonance, the dual-mode resonator side length is found to be equal to 13.52 mm with the ratios of w ₁/L ₀ and w ₂/L ₀ of about 0.1 and 0.34, respectively.

Based on this filter, two other filters with their ground planes being defected with CSRRs fractal structures corresponding to those depicted in Figs. 2(b) and 2(c) have been modeled using the prescribed substrate. As these structures imply, the inner split rings are in the form of Minkowski-like fractal geometry of the first and second iterations, respectively. In both cases, the dimensions of the CSRR are such that the side length of the outer ring is approximately equal to that of the dual-mode resonator. The corresponding inner fractal-shaped rings are scaled such that the inter-ring separations between the outer and the inner rings are found to be 0.65 and 0.37 mm to achieve the required coupling. The gaps in the two split rings, for the two iterations, have been made equal to about 0.65 and 0.55 mm, respectively to reach the specified performance.

IV. PERFORMANCE EVALUATION

The filter structure, depicted in Fig. 2(a) with no CSRR in its ground plane, has been modeled and analyzed at the design frequency, using the commercially available EM simulator, CST Microwave Studio [34]. The dual-mode resonator side length is found to be equal to 13.52 mm at resonance. This length represents about 0.11 the guided wavelength, λ_g, which is given by

(2)$${\lambda _g}=\displaystyle{{{\lambda _0}} \over {\sqrt {{\varepsilon _{eff}}} }}\comma \;$$

where ε_eff is the effective dielectric constant and can be calculated by empirical expressions reported in the literature [Reference Pozar35]. For the present case, λ_g has been found to be 89.35 mm at the design frequency. Figure 3 shows a photo of a prototype of this filter fabricated with mentioned dimensions and substrate parameters. Measured and simulated return loss and transmission responses offered by this filter are shown in Fig. 4.

Fig. 3.

A photo of the dual-mode microstrip filter prototype based on Fig. 2(a) without CSRR DGS.

Fig. 4.

Measured and simulated return loss and transmission responses of the filter shown in Fig. 3.

The bottom right of the fractal-based filter depicted in Fig. 3 has been modified to introduce a dissymmetry in the resonator structure. This dissymmetry is important because it makes possible controlling the coupling between the filter two orthogonal modes [Reference Ali23]. This topology introduces two transmission zeros, one on each side of the passband; enhancing the selectivity of the filter as shown in Fig. 4. This filter is therefore a second order with two transmission zeros. The equivalent circuit of this filter is composed of two similar LC sections connected in cascade; with input and output couplings. Each LC section represents a resonant mode of the dual-mode filter. The input coupling produces the transmission zero on the left side of the passband, whereas the output coupling creates the transmission zero on the right side of the passband. The positions of the transmission zeros can be controlled by varying the input/output coupling. The transmission zero frequency on the left side of the passband can be controlled by varying the input coupling without changing the position of the transmission zero on the right side of the passband and vice versa.

Results shown in Fig. 4 imply that the filter offers good return loss response with an in band level of more than −20 dB, and a transmission response with two transmission zeros almost symmetrically located around the design frequency and an out of the band response of more than −25 dB. In addition, measured and simulated results are in good agreement.

Moreover, two other BPF filters have been designed; both with the same dual-mode resonator depicted in Fig. 3. The first filter has the CSRR DGS structure based on the first iteration Minkowski-like fractal geometry as shown in Fig. 2(b), whereas the other filter has the CSRR DGS shown in Fig. 2(c) which is based on the second iteration of the same fractal geometry. Figure 2 demonstrates the dimensions of the simulated filters. Detailed dimensions of the dual-mode resonator, which is the same for the three filters, are depicted in Fig. 2(a). The details of CSRRs of the filters with DGS of the first and second iterations are shown in Figs. 2(b) and 2(c), respectively.

In this context, the topology of the introduced CSRR DGS has an equivalent circuit model composed of parallel combination of the ring inductance L _r and ring capacitance C _r in series with a coupling capacitance C _r [Reference Wu, Mu, Dai and Jiao19]. It is expected then, that the filter structure will have a third transmission zero. The position of this transmission zero is given by the frequency that nulls the shunt impedance as follows:

(3)$${f_z}=\displaystyle{1 \over {2\pi \sqrt {{L_r}\lpar {C_r}+{C_c}\rpar } }}.$$

The prototype photo of the first filter is shown in Fig. 5. The corresponding measured and simulated results of return loss and transmission responses offered by this filters are shown in Fig. 6. As it is implied in this figure, the upper resonant band which is attributed by the dual-mode structure is maintained; as if this resonator performs alone regardless of the existence of the CSRR DGS. The effect of the embedded CSRR DGS is assured by the appearance of the lower resonant bands. There is no such a resonant band associated with the response of the filter with non-defected ground plane. Higher fractal iteration level of the CSRR leads reduced lower resonant frequency, and consequently, makes the filter gains further miniaturization or, in other word, provides the designer with a practically useful means to tune the resulting filter response to the specified frequencies.

Fig. 5.

Photos of the fabricated filter prototype with its CSRR DGS structure is based on the first iteration Minkowski-like fractal geometry: (a) top view and (b) bottom view.

Fig. 6.

Measured and simulated return loss and transmission responses of the filter depicted in Fig. 5.

This is confirmed by the other filter shown in Fig. 7, where its CSRR DGS structure is based on the second iteration Minkowski-like fractal geometry. The corresponding measured and simulated return loss and transmission responses of this filter are shown in Fig. 8. Here, it has been noted again that the upper resonant band is almost maintained unchanged, as compared with those observed in Figs. 4 and 6 for the filters without CSRR DGS and first iteration fractal-based CSRR DGS shown in Figs. 3 and 5, respectively. On the other hand, the lower resonant frequency of this filter is lower than that produced by the filters with first iteration fractal-based CSRR DGS shown in Fig. 5. This is, of course, due to the extra length provided by the second iteration fractal structure compared with that provided by the CSRR DGS based on the first iteration fractal structure, as equation (2) implies.

Fig. 7.

Photos of the fabricated filter prototype with its CSRR DGS structure is based on the second iteration Minkowski-like fractal geometry: (a) top view and (b) bottom view.

Fig. 8.

Measured and simulated return loss and transmission responses of the filter shown in Fig. 7.

In this respect, more miniaturization could be obtained when using higher iteration levels as long as fabrication tolerances permit [Reference Ali and Hussain36]. Furthermore, as Fig. 1 implies, varying the ratios of w ₁/L ₀ and w ₂/L ₀ will lead to different values of the inner split-ring perimeters, which in turn result in a wide range of the lower resonant frequencies. The lower resonating bands as a result of the CSRRs depicted in Figs. 2(b) and 2(c) are 1.81 and 1.45 GHz, respectively. These are equivalent to size reductions, in terms of the corresponding guided wavelengths, of about 0.12 and 0.09λ_g, respectively. For the sake of comparison, the measured and simulated return loss and transmission responses of the three presented filters are summarized in Figs. 9 and 10, respectively.

Fig. 9.

Summary of the measured and simulated return loss responses of the filters shown in Figs. 3, 5, and 7.

Fig. 10.

Summary of the measured and simulated transmission responses of the filters shown in Figs. 3, 5, and 7.

The ratio of the lower resonant frequencies attributed by the split rings depicted in Figs. 2(b) and 2(c) is equal to 1.25 which is very close to the ratio of their corresponding perimeters, 1.42 as calculated using equation (1). This ratio becomes closer, 1.33, if the slight change in the guided wavelength is to be taken into account. This interesting result indicates that increasing the perimeter of the inner split ring, due to the higher fractal iteration level, results in lowering the CSRR resonant frequency. Based on this, for different values of w ₁ and w ₂, Fig. 3(a), a wide range of the inner split-ring perimeters will result in. It is then expected to design dual-band filters with fixed higher frequency and wide range of lower resonant frequencies. Consequently, this indicates that using inner split rings with structures based on fractal geometries having higher space-filling properties such as Hilbert and Peano fractal curves will result in lower resonant frequencies.

Figure 11 shows the surface current distribution on the top and the bottom of the filter structure at the centers of the two passbands. It is clear again that the two resonating bands are independent of each other; in Fig. 11(a), the surface current density on the surface of the dual-mode structure is high at the upper band, whereas it approaches zero at the lower band. The reverse is true in Fig. 11(b), for the surface current density on the bottom ground plane. It is worth to note that the same color scale is adopted for all figures.

Fig. 11.

Simulated surface current distributions at the lower and the upper passbands on (a) the dual-mode ring on the top structure and (b) bottom of the ground plane containing the CSRR.