диафрагмированные волноводные фильтры / c430ad30-9501-4330-b750-cda1e6afda7a

Abstract — This paper presents a brand-new method and process of building a broadband bandpass filter in a flat rectangular waveguide with a passband between 6-12GHz (66.7% in percentage) using a 3-layer periodic SRR structure. Due to the unsatisfied reflection in the passband, another scale of SRRs have been utilized as the lambda/4 impedance matching layer between the air and the metamaterial filter layer in the waveguide in order to decrease the reflection coefficient which is lower than -10dB after implement.

Index Terms — Metamaterial, Waveguide Filter, Split Ring Resonator (SRR), Periodic Structure, Impedance Matching Layer (IML), Flat Waveguide.

I. INTRODUCTION

Metamaterial is a conceptual material that was come up two decades ago. It represents series kinds of small periodic electromagnetic structures or its unit cells that could change the characteristics of the incident electromagnetic waves such as propagation direction, wave number, phase velocity and etc.[1] The basic principle of metamaterial is resonation, which can be easily understood. But the near field calculation around or inside metamaterial structure is quite complex. In this case, a method called effective medium theory has been proposed to analyze the far field characteristics of metamaterial without concerning about the details of electromagnetic field near or inside the periodic structures. In this way, the designing process becomes much simpler by considering the complicated repeated structures as homogeneous media with constant permittivity and permeability. The only thing researchers need to do before designing the entire structure is to measure those two constitutive parameters by classical effective parameter determination methods like Smith’s method[2].

In our research, we first investigate the single layer SRR structure, which exhibits a narrow passband and a narrow stopband close to each other. However, we note that when multiple layers of SRR are applied, the bandwidth can be expanded obviously. Next, we insert the designed SRR structure into the waveguide to evaluate the performance of metamaterial in specific situation. At last, we discovered that it is necessary to insert an impedance matching layer (IML) between the air and the metamaterial filter layer. Surely we would make use of SRRs of another scale to implement the IML.

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(1) e-mail: yanliu.seu@gmail.com

II. THEORY ANALYSIS AND DESIGN

In this section, we first start from discussing about the basic circuit principles and then analyze the metamaterial implemented in waveguide. Then, the method on how to add the impedance matching layer would be stressed.

A. Equivalent Circuits Analysis

Considering a parallel LC resonant circuit as shown in Fig. 1(a), a parallel LC resonant unit is inserted into a circuit in the series way. And the effective circuit at the resonant frequency point will be an open circuit due to the reasons below.

Open

(a) (b)

Figure 1. (a) Parallel LC resonance. (b) Its effective circuit at the resonating frequency point.

Assume that inductance L and capacitance C are utilized in the circuit in the way metioned above and the total impedance Z is

means an open circuit equivalently.

Split Ring Resonator is a kind of classical metamaterial unit structure as shown in Fig. 2(a), in which the lighter colored part is the dielectric-slab and the darker colored part is the metal attached to the slab[3]. Under the polarization of incident electromagnetic waves as illustrated in Fig. 2(b), the SRR can be simplified to an equivalent circuit[4] as shown Fig. 2(c). Due to the magnetic field would take a dominate role over electric field for resonation by passing through the loop of SRR, there will be a varying induced current along the metal lines. On the other hand, the electric lines are parallel to the coupling metal lines which have the length of l. Thus, the effective capacitance results from E-field can be ignored in this situation. So, it is clear that SRR can be seen as a resonator consists of an inductance and a capacitance with head-tail connecting under the condition of the incident electromagnetic waves shown in Fig. 2(b).

g

E

Figure 2. (a) The SRR structure. (b) The polarization of incident EM wave. (c) The equivalent circuit model.

Since the equivalent circuit of single SRR unit cell is a parallel LC resonator, we can apply multiply layers of SRR to build a lowpass filter that has a passband of 12GHz. Due to the highpass characteristic of metal waveguide, it is possible to build a 6GHz-to-12GHz bandpass filter with a combination of both waveguide and multiply layers of SRR. According to the automatic filter design software Filter Solutions, we can easily achieve the elliptic lowpass filter in the implement of lumped elements. We plan to build the filter with three elements of SRR in the propagation direction for better performance and acceptable size. The lumped elements circuit represent a simplified symmetry filter based on a 7th order lowpass elliptic filter (with the passband of 12GHz, the stopband of 12.5GHz and the passband ripple of 0.1dB) shown in Fig. 3(a). And its S parameters are displayed in Fig. 3(b).

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(a)

(b)

Figure 3. (a) The lumped elements implemented filter based on elliptic lowpass filter with passband of 12GHz, stopband of 12.5GHz and passband ripple of 0.1dB. (b) Its S parameters.

Figure 4. The periodic slabs of SRR elements. There are three elements in the propagation direction z.

And the periodic structure of SRR implemented to the filter is shown in Fig. 4. From the construction picture, the propagation direction is parallel to the positive direction of z axis in which three SRR elements are arranged. Clearly, electric field and magnetic field are varying along y and x axes respectively. As already analyzed in the paragraph of single SRR unit cell, the magnetic field penetrates through the loops perpendicularly causing surface currents especially on the resonant frequency point. On the opposite, the electric field influences slightly on the single SRR but affects largely between SRRs in the y axis direction due to the capacitance composed by two close and parallel metal lines of two adjacent SRRs on the substrate. What’s more, the boundaries of the y direction would be considered as electric walls while those in the x direction are considered as magnetic walls to make sure that the incident wave would be pure plane wave for more accurate analysis on the periodic metamaterial (each SRR element of the structure would have no difference under the effect of plane wave and these boundary conditions).

B. Analysis in Waveguide

We will analyze the characteristics of the metamaterial slabs after being inserted into the rectangular waveguide. In order to suppress high-order mode and preserve signal integrity, the waveguide should work only in the main mode which quietly determines the range of the waveguide’s size. For a rectangular waveguide, the cut-off frequency of mode TE20 should be designed higher than the passband frequency, 12GHz; and that of the main mode (TE10) should be lower than 6GHz.

That is,

(2)

√( ) ( )

where a and b is the width and height of the rectangular waveguide. When m=1, n=0,

;

when m=2, n=0,

.

Thus a must be 25 mm.

To form an electric field that is better parallel to y axis and make it no difference to affect each SRR cell in the structure (also for a better connection to circular waveguide), we suggest make use of a flat rectangular waveguide in which there is only one element in the direction of y axis. In this case, the effective shunt capacitance would be simply determined by the distance between the loops and the waveguide walls which is parallel to H-plane. Or, in other words, it is determined by the size of the SRR elements.

Although the electromagnetic waves transmitted through the waveguide are definitely not TEM waves, the disadvantageous influence is slight in the center of the waveguide. Only in the area close to the waveguide walls which is parallel to E- plane, the filtering effect will be reduced (less frequency selectivity) because the magnetic field lines do not pass through the SRR loops perpendicularly.

In the condition that has been analyzed above, the filter’s efficiency would be lower but acceptable in the waveguide.

where lm is the length of the lambda/4 IML.

related to the number of elements in the z direction could also be calculated by equation (7).

The structure, the size and the number of unit cell in the impedance matching layer are all fixed, so the design of the IML is accomplished. The process will be utilized in details in the next section.

wavelength equation of mode TE10 in the rectangular waveguide, the wavelength in the IML inserted into the waveguide is

And then the structure is inserted into the flat rectangular waveguide with the width and height

(single SRR element in the y direction). The structure is illustrated in Fig. 6(a) and the S parameters (port 1 and 2 are connected to two ends of the flat waveguide) are displayed in Fig.6(b). It clearly shows that the filter has a passband of 6.7- 12.5GHz with a maximum reflection of -8.4dB at approxi-

mately 11GHz. Except two narrow unwanted passband at 14.5GHz and 18.3GHz, every parameter is acceptable.

tion of -10dB. There are seldom papers studying on metamaterial for waveguide filters before. And most of classical rectangular waveguide filters are built to be narrow band. Thus, the combination of waveguide and metamaterial provides a new method to build broadband bandpass waveguide filters. And it is even more easily to build than the traditional waveguide building approaches.

However, a few unwanted characteristics are still influencing the performance of the flat rectangular waveguide filter. The study on problem analysis and optimization should still in process.

(b)

Figure 6. (a) Simulated structure. (b) S parameters.

The effective constitutive parameters of the filter layer can be obtained from the simulated S parameters of the single SRR element in the same size. The relative wave impedance of the periodically structured SRR element is obtained to be 0.9101 at the frequency 11GHz (due to the largest reflec-

SRR elements implemented in the z direction for impedance matching (with a period of 3mm in z direction) is approximately 2.

With slightly optimizing ( ), the final S parameters are shown in Fig. 8. It is clear that the filter has a passband of 6.2-12.1GHz with a maximum reflection of -10dB. Except two narrow unwanted passband at 14GHz and 18.3GHz, every parameter is satisfied.

IV. CONCLUSION

This paper presents a brand-new method of building a broadband bandpass filter in a flat rectangular waveguide using periodic metamaterial structure of SRRs. The filter has a passband of 6.2-12.1GHz (64.48%) with a maximum reflec-

Figure 7. The impedance of the matching SRR elements with different g.

Figure 8. The S parameters of the flat rectangular waveguide filter with an IML

REFERENCES

[1]V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys.-Usp., vol. 10, no.4, pp.509-514, Ja n.-Feb. 1968.

[2]D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B, vol. 65 no. 19, pp. 195104, 2002

[3]X. Chen, “Design and Experiment of Metamaterial Slab Lens Antennas,”

Dissertation of MEng. In Southeast Univ.(China), pp. 5-7, 2009.

[4]J. D. Baena, J. Bonache, F. Martín, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcí–García, I. Gil, M. F. Portillo and M. Sorolla, “Equivalent-Circuit Models for Split-Ring Resonators and Complementary Split-Ring Resonators Coupled to Planar Transmission Lines,” IEEE Trans. on microwave theory and techniques, vol. 53, no. 4, pp. 1451-1461, April 2005.