диафрагмированные волноводные фильтры / e508035a-4e8b-44cf-b444-7b9653f341b6

Abstract—The design and optimization of low insertion loss finline filters is described in this paper. Finline filter with a large pass band has been designed using equivalent circuit approach. Mode matching method has been used for the analysis of discontinuities in the filter. The performance of the filter obtained through this method is further optimized to achieve improved pass band and stop-band performance. A practical Quasi-Newton algorithm has been used for this purpose.

Keywords—Finline filter, Mode matching method.

I.INTRODUCTION

Finline and E-plane filters have been widely used in microwave and millimeter wave applications and have been reported extensively [1-5]. A finline is obtained from a finite thickness of metal on a dielectric substrate. A bandpass filter in finline technology is low-cost and easy to fabricate. Several design variants of finline filters for narrow band applications have been discussed by Vahldieck [1]. The initial design of the filters were based on equivalent circuit approach after modeling the discontinuities of finline strips as inductive elements. The designed filters were further numerically analyzed by solving the field theory problem at the discontinuities using Mode Matching Method(MMM) and subsequently optimized to achieve accurate specifications. Commercially available 3- D field solvers have been used recently in the design and optimization of finline filters. Design and optimization E-plane filter using finline posts as well as inductive finline strips for broadband applications has been reported by Xu et al [6]. Such a filter with finline posts was achieved improved stop-band characteristic for a small increase in the length of the filter in comparison to that with inductive strips. A broadband three coupled finline filter was proposed by Rao and Rao [7]. The structure described by Rao and Rao includes a tapered finline structure as well as the coupled finlines. This paper presents a finline filter with thin inductive strips designed for a large bandwidth of ten percent in the X-band. Analysis of the filter discontinuities is based on MMM. The initial design is arrived from the equivalent theory approach. The filter structure is further optimized for improved performance in the pass band and stop-band using a practical Quasi-Newton algorithm as presented in [8].

II.THEORY

The design procedure of a finline filter or any other type of filter begins with network synthesis based on the performance requirements. In this paper the design a finline bandpass filter for ten percent bandwidth in X band with a center frequency of 10.4GHz and insertion loss of less than 0.1dB in the passband is presented. A fifth order Chebyshev filter response

was selected for the design to provide sufficient attenuation in the stop band. The lumped elements of low-pass prototype with unity cutoff for this response from which this filter is synthesized is obtained as g1 = 1:301; g2 = 1:556; g3 = 2:241; g4 = 1:556; g5 = 1:301 [2].

The g1, g3 and g5 represents the shunt capacitors while g4 and g5 the series inductors of the lumped element ladder network of the low-pass filter prototype with unity cutoff and normalized source and load resistance. The bandpass filter is obtained using suitable frequency transformation. The transformation converts inductors of low-pass filter to series resonant circuits and capacitors to parallel resonant circuits. The series resonant circuit in the finline topology is formed by half-wave length waveguide sections as shown in Figure 1. Parallel resonant circuits which are shunt elements are realized in waveguide technology using impedance inverters K as shown in Figure 2. In the case of finline filters the impedance inverters are realized using finite length of finline bifurcations. They transform the series-connected elements to shunt-connected elements. Based on the low-pass to bandpass transformation and the equivalent circuit for half wavelength waveguide resonator sections, the values of the impedance inverters shown in Figure 2 are obtained from the equations below. The procedure to obtain these equations are described in [5].

where, w is the guide wavelength fractional bandwidth of the filter. It is basically the difference in the guide wavelength at the pass band edges over the guide wavelength at the center frequency of operation of the filter.

The equivalent circuit of the finite lengths of the sections of finline waveguide bifurcation [3] that perform the impedance transformation is shown in Figure 3. It comprises of a T network of inductive reactances and short lengths of transmission line at either ends. The reactances of the circuit are obtained from the scattering parameter of the finite length finline bifurcation as given below. The procedure to obtain the S-parameters of the bifurcation is described later.

The impedance inverter ’K’ values and the lengths of the transmission line =2 in the equivalent circuit of the bifurcation are related to the reactances of equivalent network and are given below [3].

The lengths of the transmission line in the equivalent circuit of the finline bifurcation as obtained from the above equation are negative. However, they are absorbed into the lengths lj of waveguide sections that form the resonators and are adjacent to the two neighboring lines j and j+1. The electrical length j of the resonators which are or half the guide wavelength at the center frequency of pass band of the filter now include these additional line lengths and hence related to the physical lengths lj as below.

The procedure in designing the filter would be to first determine the value of impedance inverters K from the g elements for the desired performance and fractional bandwidth of the filter. These K values are realized by determining the lengths of the finline bifurcation that provide them using equations (4) through (7). The mode matching method of analysis of the discontinuity is used to obtain the S-parameters for varying finite lengths of the finline bifurcation is used for this purpose. A set of normalized reactance’s versus length of finline sections has been presented in [1] for selected thickness of the finline. However, for the filter presented here, the lengths of the finline bifurcation for initial design have been obtained using the mode matching analysis presented below. The resonator lengths lj have been calculated by setting j = in equation (8).

To obtain the S-parameters of the finline sections using mode matching method the fields on both sides of the discontinuity are expanded in terms of a series of modes of incident and reflected waves.The region I of the discontinuity as shown in Figure 1 is an empty waveguide section. The inductive strip of finline in the waveguide has been shown to have three regions; the region II is the dielectric filled region of the finline strip and region III and IV are the empty waveguide sections. The thickness of the metal on the finline is finite and included in the analysis. This type of discontinuity is in the E-plane only, and hence when a T E10 mode is incident in the empty waveguide section, field components other than Ey, Hx and Hz are not introduced. Therefore, the fields at such a discontinuity can be described by either T Emx 0 or T Emz 0 modes as they represent the same field configurations. The fields of T Emx 0 modes are derived from vector potential Ahx and is used here to derive the fields present in the empty waveguide sections and dielectric filled waveguide sections.

The vector potentials in each of the region ’r’ of the discontinuity are written as sum of the incident and the

reflected normal modes as given below,

The coefficients Cm(r) represent the power normalization constants and are obtained by setting the magnitude of the power carried in each of the modes to unity. The term p(r)=q(r) is written so that the potential function which is a solution of the vector Helmholtz equation satisfies the boundary conditions in the region (r) and is given in the Appendix. The relationship between the coefficients Fm(r) and Bm(r) (i.e. the incident and reflected wave amplitude) that will be obtained from the analysis gives the generalized scattering matrix of the discontinuity.

The next step in MMM is to match the tangential component of electric and magnetic fields at the interface of the discontinuity. In such a discontinuity as shown in Figure 1, it is sufficient to match Ey and Hx components to obtain the S-parameter of the discontinuity. The matching condition is written as,

EyI = EyIV + EyII + EyIII in Region IV,II, III

=0 u t=2 < x < u + t=2, v t=2 < x < v + t=2

Using the above matching conditions, together with the principle of orthogonality of the modes the following sets of algebraic equations are obtained,

(F I + BI) = [LII](F II + BII) +

where Lr are the coupling matrices that relate the coefficients of the incident and reflected waves at the discontinuity and are as given in the appendix. The matrices [Lr]0 are the transpose of the coupling matrices Lr. The size of these matrices are determined by the number of modes incorporated in the analysis of the discontinuity. Algebraic manipulations of the above equations result in the generalized scattering matrix of the discontinuity from empty waveguide to a finline. However the bandpass filter is composed of finite lengths of the finline section and empty waveguide resonator sections. In order to obtain the generalized scattering matrix of a finite length of finline section, algebraic manipulations are done on the above equations and four more such equations written for the discontinuity from finline section to waveguide as shown in Figure 1. Since the coefficients F rL and BrL of each of the regions are related to F r and Br by the propagation constants times the length of the finline section the generalized scattering matrix of such a discontinuity is obtained and as given in the appendix. The S-parameter of the fundamental mode is

extracted from the generalized scattering matrix for the design process. The initial dimensions of the filter are thus obtained using this approach.

All the discontinuities that form the filter are analysed by MMM and their generalized scattering matrix are cascaded to obtain the performance of the filter. A practical QuasiNewton algorithm [8] was used to further optimize the filter dimension. The performance of filter at selected frequency points in the pass band and stop band was utilized to minimize the error function for the optimization process. The error that is minimized is obtained from the filters reflection coefficient at the selected frequency points fm in the stop band and transmission coefficient at the selected frequency points fi in the pass band and is as follows,

MI

The total number of selected frequency points was about fifty for the fifth order filter that has been designed. The optimized dimension and the performance is discussed in the next section.

III.RESULTS

The Chebyshev filter was designed and optimized for improved pass band and stop band characteristic and its response is shown in Figure 4. The mode matching analysis included twenty five modes to obtain convergence of the S-parameters. A comparison of this filter with the three coupled finline filter [7] is made here. The dimension of the filter in [7] is 61.7mm while the dimension of the filter obtained here is 57.9mm for similar but not identical specifications of the two filter.

IV. CONCLUSION

This paper describes computer aided design of finline filters using mode matching method and a practical quasiNewton optimization for obtaining improved performance. Recently broadband finline filters utilizing several types of discontinuities such as posts or three coupled structure have been described. Although several types of discontinuities could be used in design of finline filters to improve the performance of the filter, it is concluded that a comparison with traditional structure is necessary to indicate if any improvement can be achieved through the proposed structure. The finline filter presented using conventional approach provides comparable performance to that of the three coupled structure.

APPENDIX

The functions satisfying the boundary conditions of the vector potential solutions in each of the region is given by,

23

The matrix containing the coupling integrals involved in the determination of S-matrix of a discontinuity from empty waveguide section to a finline is given by

The submatrices of the generalized S-matrix of an empty waveguide to finite length of finline section is given as

where

IV

P= U + X LrDIIXrDrLr0

r=II

IV

Q = 2U + X LrXr[Lr]0

r=II

IV

R= 2 X LrXrDr[Lr]0

r=II

where

Dr = Diag[e jkzir lf ]

X= (U DrDr) 1

where Diag represents a diagonal matrix. D is the diagonal matrix of the exponential of the propagation constants of various modes times the length of the finline section(lf ) and U in all of the above equations is an identity matrix of appropriate size.

REFERENCES

[1]Ruediger Vahldieck, Quasi Planar Filters for Millimeter-wave Applications, IEEE Trans. Microwave Theory and Techniques, Vol 37, No.2, February 1989, pp. 324-334

[2]G.L. Matthaei, L. Young and E.M.T. Jones, Microwave filters, Impedance-matching networks and coupling structures, McGraw-Hill, 1964.

Fig. 1. Finline filter and its discontinuity regions

Fig. 2. Impedance inverter within the bandpass filter structure

[3]Y. Konishi and K. Uenakada, ” The design of bandpass filter with inductive strip - Planar circuit mounted in waveguide”, IEEE Transactions of Microwave Theory & Techniques, Vol.22, No. 10, pp. 869-873, Oct 1974.

[4]Y. C. Shih, T. Itoh, and L. Q. Bui, Computer-aided design of millimeterwave E-plane filters, IEEE Trans. Microwave Theory Tech., vol.31, no. 2, pp. 135142, Feb. 1983.

[5]J. Uher, J. Bornemann and U. Rosengerg, Waveguide Components for antenna feed systems: Theory and CAD, Artech House Inc., 1993.

[6]Zhengbin Xu, Jian Guo, Cheng Qian, and Wenbin Dou, Broad-Band E- plane Filters With Improved Stop-Band Performance, IEEE Microwave and Wireless Components Letters, Vol. 21, No. 7, July 2011, pp. 350-352.

[7]V. Madhusudana Rao and B. Prabhakara Rao, Design of Three-Coupled Finline Bandpass Filters with Full Wave Analysis, Progress In Electromagnetics Research Symposium Proceedings, Taipei, March 25-28, 2013, 895-899.

[8]A. Antoniou, Digital Filters, Design and Applications, Second Edition, McGraw-Hill, 1993.

Fig. 3. Equivalent circuit of the finline bifurcation

Frequency in GHz

Fig. 4. Characteristic of a five resonator optimized finline filter, waveguide dimensions a=22.86mm, b=10.16mm, thickness of the dielectric substrate=0.8mm, r=2.22,t=35.56 m, total length of the filter =57.9mm, length of the first and sixth finnline section=0.7mm, length of the second and fifth finline section=0.9mm, length of third finline section =0.7mm, length of the first and fifth waveguide resonator=1.35cm,length of the second and fourth resonator=0.67cm, length of the third resonator = 1.3cm