диафрагмированные волноводные фильтры / 8cfb3b12-64e4-41d9-bccb-3a7cb6861c22

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13.M.K. Chin and W.S.C. Chang, Theoretical design optimization of multiple-quantum-well electroabsorption waveguide modulators, IEEE J Quant Electron 29 (1993), 2476 –2487.

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15.M.P. Pires, B. Yavich, and P.L. Souza, Chirp dependence in InGaAs/ InAlAs multiple quantum well electro-absorptive modulators near polarization-independent conditions, Appl Phys Lett 75 (1999), 271– 273.

©2004 Wiley Periodicals, Inc.

ANALYSIS OF WAVEGUIDE DISCONTINUITIES USING A HYBRID METHOD COMBINING THE HELMHOLTZ WEAK FORM AND MODE EXPANSION

H. C. Wu and W. B. Dou

State Key Laboratory of Millimeter Waves

Southeast University

Nanjing 210096, P. R. China

Received 19 March 2004

ABSTRACT: A new approach for the analysis of waveguide multiports discontinuities is presented in this paper. This approach is a combined method of the mode-expansion method and finite-element method, which is based on the weak form of the Helmholtz equation. We use a mode expansion technique other than PML to truncate the computing space of finite element. Multiport structures such as H-Plane forked T-junction, waveguide loaded with two dielectric posts, and E-plane filter are studied as examples. The calculated results of these structures are shown and compared, respectively, to those in references. Good agreement between them is observed. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 43: 173–177, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20411

Key words: weak form of Helmholtz equation; finite-element methods; mode expansion; multi-port discontinuities

1. INTRODUCTION

Approaches often used to analyze waveguide discontinuities include MMM, MoM, FEM, BEM, FDTD, and so on. The MMM can be used to analyze regular-shape structures, but it has great difficulty in dealing with problems such as complicated boundary conditions. The MoM can be used to analyze structures with complicated boundary conditions, however, the matrix of coefficients obtained based on integral equations is usually a filled matrix and each element of the matrix occupies some computation time. Thus, a great deal of computation time is needed for this

method. Also, during the computing process of the matrix element, singularity needs to be dealt with carefully. The BEM also has the disadvantage of needing to resolve the problem of singularity. FDTD is suitable for analyzing the structure whose shape coincides with the shape of the grid. As the staircase technique is used to simulate the irregular boundary, the results obtained by this method may not be accurate. The classical FEM is based upon variational formulation; thus, the operator should be self-adjoint, hence different variational formulation is needed for different problem. For example, a generalized variational formulation based on different inner products is needed for problems containing loss dielectrics, and extremum procedure is needed to get final matrix equation [1]. In some cases, the variational formulation may be also difficult to obtain.

Based on the concept of a weak solution of a partial differential equation, Peterson introduced the Helmholtz weak form and analyzed some two-dimensional (2D) problems of electromagnetic scattering from inhomogeneous cylinders [2]. This approach may originate from the concept of a weak solution of partial differential equations. The solution of the Helmholtz weak form is the weak solution of the Helmholtz equation. Based on the Helmholtz weak form, Arena proposed a new method (a hybrid mode-matching/ finite-element approach) in 2001 [3] to analyze two-port waveguide discontinuities, in which the inner part is regular and isotropic. However, this method is only limited to treat multidiscontinuities with the same waveguide cross-plane changing along the propagation direction. In a previous paper, a different hybrid method that combines finite-element analysis based on the weak form of Helmholtz equation and field matching, which has been used to analyze waveguide T-junction and bend, was presented [4]. In this paper, this method is used to deal with multiport waveguide discontinuities with complex boundary and containing full-height posts. Based on this weak form, by using interpolation to present the electric and magnetic fields in the complicated structure and combining the mode expansion in the regular region in order to truncate the computing region, we can finally obtain a set of linear equations of the problem. In this paper, structures such as the forked T-junction, waveguide loaded with two square dielectric posts, and E-plane filter are studied as examples to verify the validity and accuracy of the method.

2. APPROACH

2.1. Helmholtz Weak Form

In the inhomogeneous, anisotropic passive media, Maxwell’s equations may be written as

Substituting Eq. (4) into Eq. (2), we obtain

(9)

2.2. Hybrid Mode-Matching and Finite-Element Methods

We take the T-junction as an example, as shown in Figure 1. Parameter a is the width of the waveguide.

(10)In the junction region I:

l 1

whereas here, Bn( x, y) is the linear pyramid interpolation function

in finite-element method. And esizl( x, y) (i 1, 2, 3) are the waveguide modes given as follows:

Let T Bi( x, y) i 1, 2, . . . , N in the Helmholtz weak form:

By multiplying sin( y a/ 2)(m /a) on both sides of Eq. (31) below, and integrating from [ (a/2), (a/2)] for y, we can obtain L linear equations.

Figure 2 Forked T-junction

Figure 3 Performance of the forked T-junction

Also, multiplying sin( x a/ 2)(m /a) on both sides of Eqs. (32) and (33) below, and integrating from [ (a/ 2), (a/ 2)] for x, respectively, we can obtain another 2 L linear equations.

Now we finally obtain:

AX B

A: N 3L N 3L , B: N 3L 1, X: N 3L 1

X A1, A2, . . . , AN, es11, e12s , . . . , e1s L, es21,

es22, . . . , es2L, es31, es32, . . . , es3L . (31)

Thus, the S-parameters can be obtained by the following equations:

Figure 4 Waveguide loaded with two dielectric posts

Figure 5 Performance of the forked T-junction ( fc is the cutoff frequency of the waveguide)

3. NUMERICAL RESULTS AND DISCUSSIONS

3.1. Numerical Results for Forked T-junction

Forked T-junction is often needed in a compact microwave system for power dividing. In Figure 2 an example of such T-junction is shown. Figure 3 shows the calculated parameters S11 and S21. S31 S21 because of the symmetry of the structure. It is also compared with the results from FDTD method. Because the grid of FEM can be made to be fine in the corner region it is believed that the calculation results of FEM is more precision, though two results have better consistency.

3.2.Numerical Results for the Waveguide Loaded with Two Square Dielectric Posts

Waveguide loaded with two dielectric posts as shown in Figure 4 can be used as filter in microwave systems. It is analyzed as second example. To deal with it is simple. Figure 5 shows its performance, as expected it has property of filter. The calculation results is compared with that of FDTD, good agreement has been obtained because the structure is regular.

3.3.Numerical Results for the E-plane Filter

Finally, waveguide loaded with conducting post as shown in Figure 6, which is also called E-plane filter, is analyzed. The grid near the corner of the post should be finer to express the field more precision and the number of grid should be increased to obtain a convergence results. Calculation results are depicted in Figure 7 and compared with that from ref. [7], two results have good agreement.

4. CONCLUSION

Waveguide discontinuities with complicated structures can be analyzed by the new method proposed this paper. Some examples,

Figure 6 E-plane filter

Figure 7 Performance of the E-plane filter

including forked T-junction, waveguide loaded with two square dielectric post, and E-plane filters, have been analyzed. The results have been compared to those in references, and good agreement between them has been observed. The computation error of energy conservation is less than 10 6. Thus, we can make the conclusion that the new method proposed in this paper is correct.

This method has the following characteristics:

●As it is based on the Helmholtz weak form, the operator does not need to be self-adjoint. Also, variational formulation is unnecessary to obtain. Moreover, this approach is general enough for calculating other more complex discontinuities problem.

●It weakens the restriction to the solution function. The application of the weak forms allows much wider basis functions to express the unknown field.

●It can be easily applied to analyze complicated structures. The involved computation region can be as small as possible, hence the memory space and computation time are reduced.

●The electromagnetic fields on the interface between the irregular region and regular waveguide are matched. Thus, the absorbing boundary conditions such as PML are unnecessary.

●It is easy to implement. As the matrix of the equations is sparse, this method does not require much computation time.

The analysis of all these 2D problems is only the first step of our work. This method can be extended to analyze 3D problems and the studies on this will be presented in the near future.

REFERENCES

1.J.M. Jin, the finite-element method of electromagnetism, Xi’dian University Press, Xi’an, 2001 (in Chinese).

2.A.F. Peterson and S.P. Castillo, A frequency-domain differential equation formulation for electromagnetic scattering from inhomogeneous cylinders, IEEE Trans AP-37 (1989), 601– 607.

3.D. Arena, M. Ludovico, G. Manara, and A. Monorchio, Analysis of waveguide discontinuities using edge elements in a hybrid mode-match- ing/finite-elements approach, IEEE Microwave Wireless Compon Lett 11 (2001), 379 –381.

4.W.B. Dou and S.H. Xu, Numerical analysis of waveguide discontinuity based on the weak forms of the Helmholtz equations, J Electromagn Waves Applic (to appear).

5.W.B. Dou and Y.Z. Zhang, Forked hybrid waveguide T-junctions analyzed by FDTD method, J Electromagn Waves Applic 17 (2003), 517–523.

6. W.B. Dou, FDTD analysis for multiple arbitrarily shaped posts in a waveguide, Microwave Opt Technol Lett 27 (2000), 216 –220.

7.Y. Shih, Design of waveguide E-plane filters with all-metal inserts, IEEE Trans Microwave Theory Tech MTT-32 (1984), 695–704.

© 2004 Wiley Periodicals, Inc.

A DUAL-MODE BANDPASS FILTER WITH A WIDE STOPBAND

Min-Hung Weng,2 Ru-Yuan Yang,1 Cheng-Yuan Hung,3

Hung-Wei Wu,3 Wu-Nan Chen,3 and Mau-Phon Houng1

1 National Nano Device Laboratories

Taiwan

2 Institute of Microelectronics

Department of Electrical Engineering

National Cheng Kung University

Tainan, Taiwan

3 Department of Computer and Communication

Shu Te University

Taiwan

Received 20 March 2004

ABSTRACT: A novel dual-mode band-pass filter with a cross-shaped resonator and three ring resonators has been designed and implemented. The proposed structure is different from the conventional dualmode band-pass filter. The designed dual-mode band-pass filter has a very sharp attenuation rate by two transmission zeros, and high spurious suppression by the band-stop effect of the three ring resonators. The experiments show good agreement with the simulations. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 43: 177–179, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20412

Key words: dual-mode band-pass filter; cross shaped resonator; attenuation; transmission zeros; spurious suppression

1. INTRODUCTION

Microwave band-pass filters with high performance and low production cost have always been required for the fast-growing wire- less-communication markets. Conventional planar filters, such as parallel-coupled half-wavelength filters, have large size, low attenuation rates, and spurious response, and therefore have not been suitable in modern communication systems [1–3]. Some of these improvements have been provided due to new type of band-pass filter structure. Dual-mode band-pass filters mainly consist of a physical coupling of two degenerate modes in a symmetrical ring or patch resonators excited by a perturbation element, such as the patch structure along the orthogonal plane of resonators. The use of dual-mode resonators to realize microwave filters has been known in recent years [2– 6].

Conventional planar dual-mode band-pass filters using halfwavelength resonators and one-wavelength resonators have spurious pass-bands at 2 f0, 3 f0, and 4 f0, where f0 is the center frequency of the band-pass filter. A high-performance filter in a sensitive receiver requires a wider upper stop-band, including 2 f0 and 3 f0 for reducing the interference from out-of-band signals.

In this paper, a novel dual-mode band-pass filter using a crossshaped resonator and three ring resonators with orthogonal input/ output (I/O) is presented. The input and output ports are spatially and symmetrically separated on ring resonators. This dual-mode band-pass filter, mainly formed by the half-wavelength crossshaped resonator, gives two degenerate modes excited by a squarepatch perturbation element. The three ring resonators of the pro-

Figure 1 Configuration of the novel dual-mode band-pass filter

posed dual-mode band-pass filter are used to generate the bandstop effect in order to suppress spurious responses and thus optimize the out-of-band performances. The designed dual-mode band-pass filter shows a novel structure and provides better spurious suppression than that of conventional shunt-stub and cou- pled-line band-pass filters. For accurate prediction, a design technique using the electromagnetic (EM) simulator in order to obtain a full-wave integral-equation solution and the method of moments (MoM) is also described.

2. NEW FILTER CONFIGURATION

Figure 1 illustrates the configuration of the novel dual-mode bandpass filter. The dual-mode band-pass filter consists of a crossshaped resonator and three ring resonators. The first basic structure of the dual-mode band-pass filter is the cross-shaped patch resonator, which is formed by orthogonally placing two arms of half-wavelength microstrip line. The two degenerate modes are excited and coupled to each other by the square-patch perturbation element within the cross-shaped patch resonator. The two modes are excited in the ring resonator and the cross-shaped resonator due to the incident power along Port 1, and coupled each other by using the square-patch perturbation element. Port 2 is orthogonal to Port 1, thus providing the output for two modes. The proposed structure includes a symmetric structure, and directed tapped I/O tapping into the ring resonators and then coupling to the cross-shaped resonator, which are different from the conventional ones that usually comprise symmetrical structure and coupling feeding. The second basic structure of the dual-mode band-pass filter is comprised of the three ring resonators. The three ring resonators of the dual-mode band-pass filter are used for producing the band-stop effect for coupling to the cross-shaped resonator in order to suppress the spurious response. Figure 2(a) shows the effect of the ring resonator upon the one arm of the cross-shaped resonator. In Figure 2(b), the simulated performance presents the ring resonator coupled to the microstrip line, which acts as band-stop element with the bandstop effect at center frequencies f0, 2f0, and 3f0. Therefore, the three ring resonators have two advantages in the design of the dual-mode band-pass filter using the cross-shaped resonator.

The perturbation formed by the square patch is attached to the corner of the cross-shaped resonator for exciting and coupling a pair of degenerate modes so as to form the dual-mode band-pass