диафрагмированные волноводные фильтры / 38571bc5-9420-4ebd-ba62-dca14d11b468

Abstract—A method to generate an accurate rational model of lossy systems from either measurements or an electromagnetic analysis is presented. The Cauchy method has been used to achieve this goal. This formulation is valid either for lossless or lossy system responses. Thus, it provides an improvement over the conventional Cauchy method and takes into account the relationship between the transmission and reflection coefficients of the system which in our case is a filter. The resulting model can be used to extract the coupling structure of the filter. Two examples have been presented. One deals with measured data and the other one uses numerical simulation data from an electromagnetic analysis.

Index Terms—Bandpass filter, Cauchy method, coupling matrix, rational model, scattering parameters.

I. INTRODUCTION

IN RECENT years the problem of generating reduced-order models from data samples obtained from an electromagnetic (EM) analysis has been an important object of research. The main goal is to reduce the number of computationally intensive EM analyzes while maintaining the accuracy of the computed response in a frequency band as wide as possible. In the case of frequency responses of passive devices, the most straightforward approach in order to generate an analytical model is to use rational polynomial interpolants [1] with the frequency as the independent variable. The Cauchy method is a convenient, in addition to accurate and fast, technique to fit rational polynomials to a specified response. The method also allows the estimation of the minimum model order and the modeling error [2]. Some effort has also been made in order to extend the Cauchy method to multidimensional functions, that is, to functions of more than one independent variable [3], [4]. However the problem presented in this letter is different and has not been previously investigated: it consists of extracting a vector function of one variable for a general system response which may be lossy. In other words, it consists of simultaneously fitting two rational functions, namely the reflection and transmission coefficients of the filter. Of course, two independent one-dimensional (1-D) interpolants can also be generated, but the transmission and reflection coefficients of a passive device are related to each other as they share a common set of poles. The approach based on two independent rational functions does not guarantee a common set of poles for a lossy system and therefore the reduced order

Manuscript received November 12, 2003; revised June 15, 2004. The review of this letter was arranged by Associate Editor J.-G. Ma.

A. G. Lampérez and M. S. Palma are with the Department of Signals, Systems and Radio Communications, Polytechnical University of Madrid, Madrid 28040, Spain.

T. K. Sarkar is with the Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13210 USA.

Digital Object Identifier 10.1109/LMWC.2004.834576

model thus generated is not a valid one. The proposed technique not only guarantees that the two rational polynomials have the same poles, but also reduces the number of data samples necessary to carry out this model order reduction. A variation of this technique has been successfully applied by the authors as part of a fast algorithm for model-based optimization [5], but in this case only the numerator polynomial coefficients of the reflection and transmission parameters have been estimated, requiring the posterior reconstruction of the common set of poles.

II. PROBLEM FORMULATION: CAUCHY METHOD

A reduced order model is constructed as a rational poly-

sense as rational polynomials are eigen-functions of a linear time invariant (LTI) system. The solution of the polynomial co-

the right null space of must have dimension greater than zero. Therefore, the model order is limited by the number of available samples, .

The total least squares method (TLS) is used to solve the matrix system in (1) which often is overdetermined. Since only the columns of are not known with absolute certainty due to measurement errors a constrained TLS problem is set up with the first columns of fixed a priori [6].

III. RATIONAL INTERPOLATORS WITH COMMON POLES

A two-port reciprocal network can be characterized by a set of two parameters of its scattering matrix, namely the reflection and transmission coefficients, and . Those parameters share the same set of poles. Therefore in order to generate a valid model of a filter response using the Cauchy method it is

equation resulting in the new system

(7)

where is the null matrix and . The system may be solved in the same way as (1) using a constrained TLS approach with fixed columns. First the common denominator coefficients are solved for and then the result is used to obtain the vector of numerator coefficients .

This algorithm has some differences with respect to another method previously presented [5]. First, the numerators and denominator are solved in one step, without requiring the reconstruction of the network poles. As a consequence, it is formulated as a TLS problem with constraints (the problem in [5] is a simple TLS one). Second, it does not impose the condition

, that is, it is not restricted to lossless networks. Hence this method is more general as it can handle lossy systems.

IV. APPLICATIONS

A. E-Plane Waveguide Filter

This new formulation dealing with the Cauchy method has been applied first to a set of measured data samples of a Ku-band bandpass filter response. The device is a sixth-order E-plane

filter built in a WR28 rectangular waveguide. The original design presents a Chebyshev-type response with a reflection parameter better than dB at the pass band. However the measured response is far from this ideal response due to low mechanical tolerances.

Fig. 1 shows the measured data and the response of a sixthorder model with no finite transmission zeros, including the mean square error (MSB) between both responses. Larger order models can not obtain further improvements. Table I contains the polynomial coefficients of the model. It should be noted that as this is a low-pass equivalent of a bandpass filter the coefficients can be complex.

The polynomials can be used to estimate the corresponding lossless ones by using the dissipation factor , that is, by shifting the roots of the polynomial by an amount to the right in the complex plane so that the dissipation loss

is minimized (ideally the minimum is zero) The lossless polynomials are the starting point of a synthesis procedure that leads to the coupling coefficients of the network [5].

B. Hairpin Microstrip Filter With Cross-Couplings

As a second example on the application of this new technique, two different models corresponding to a microstrip cross-cou- pled bandpass filter have been generated. The structure, presented in [7], is a four-pole filter formed by four identical folded hairpin resonators tuned at GHz with significant magnetic and electric coupling between each contiguous pair. The quadruplet structure with a cross coupling generates a pair of symmetrical finite transmission zeros, allowing the synthesis of an elliptic transfer function. The device has been analyzed

Fig. 2. Microstrip filter, MoM and model narrow-band response, and coupling topology (solid: main coup.; dashed: cross coup.; dotted: spurious coup.).

Fig. 3. Microstrip filter, MoM and model wide-band response, and coupling topology (solid: main coup.; dashed: cross coup.; dotted: spurious coup.).

TABLE II

MICROSTRIP FILTER: NARROW-BAND MODEL POLYNOMIAL COEFFICIENTS

using the method of moments (MoM) and both narrow-band and wide-band models have been generated using the described technique.

Fig. 2 shows the narrow-band system response obtained using MoM and the corresponding rational model. As only data samples around the pass band have been used the model can be considered a narrow-band one. The polynomials of the low-pass equivalent, whose coefficients are shown in Table II, correspond to a fourth-order filter with two finite transmission zeros. As can be seen, the location of the reflection zeros and the in-band return loss have been accurately modeled.

From this computed rational model a coupling matrix can be synthesized [8] with the topology shown in Fig. 2.

This coupling matrix reveals some features of the physical filter: first, although the resonators have been designed to be identical some frequency shifs are necessary to model the response. Second, the model is more accurate if diagonal couplings –– are allowed (their value is low).

A wide-band rational model has also been generated using data samples from both the main pass band and the first upper band (Fig. 3). In this case, two resonant modes are considered for each physical hairpin resonator, therefore the order of the filter is doubled. The resulting eight-pole structure is roughly equivalent to two four-order networks in parallel with an additional direct coupling between the source and the load. Fig. 3 shows that some second-order effects have also been modeled, i.e., the reduction of the maximum attenuation of the filter to approximately 40 dB and the shape of the upper band, including an isolated transmission zero at GHz.

V. CONCLUSION

In this letter, the Cauchy method has been presented to generate a reduced order model of systems which may be lossy. This new method fits a rational model to the numerical data samples of both the transmission and the reflection parameters of a filter response and guarantees that they have the same set of poles. Using this technique the responses from a waveguide filter and a planar filter with transmission zeros have been computed. The computed models show good agreement simultaneously with respect to both the transmission and reflection parameters, and are in correspondence with the true physical structure of the devices.

REFERENCES

[1]E. K. Miller and T. K. Sarkar, “Model-order reduction in electromagnetics using model-based parameter estimation,” in Frontiers in Electromagnetics. Piscataway, NJ: IEEE Press, 1999, pp. 371–436.

[2]R. S. Adve, T. K. Sarkar, S. M. Rao, E. K. Miller, and D. R. Pflug, “Application of the Cauchy method for extrapolating/interpolating narrow-band system responses,” IEEE Trans. Microwave Theory Tech., vol. MTT-45, pp. 837–845, May 1997.

[3]S. F. Peik, R. R. Mansour, and Y. L. Chow, “Multidimensional Cauchy method and adaptive sampling for an accurate microwave circuit modeling,” IEEE Trans. Microwave Theory Tech., vol. MTT-46, pp. 2364–2371, Dec. 1998.

[4]A. Lamecki, P. Kozakowski, and M. Mrozowski, “Efficient implementation of the Cauchy method for automated CAD-model construction,”

IEEE Microwave Wireless Compon. Lett., vol. 13, pp. 268–270, July 2003.

[5]A. García-Lamperez, S. Llorente-Romano, M. Salazar-Palma, and T. K. Sarkar, “Efficient electromagnetic optimization of microwave filters and multiplexers using rational models,” IEEE Trans. Microwave Theory Tech., vol. 52, pp. 508–521, Feb. 2004.

[6]J. W. Demmel, “The smallest perturbation of a submatrix which lowers the rank and constrained total least squares problems,” SIAM J. Numer. Anal., vol. 24, pp. 199–206, 1987.

[7]J.-T. Kuo, M.-J. Maa, and P.-H. Lu, “A microstrip elliptic function filter with compact miniaturized hairpin resonators,” IEEE Microwave Guided Wave Lett., vol. 10, pp. 94–95, Mar. 2000.

[8]R. I. Cameron, “Advanced coupling matrix synthesis techniques for microwave filters,” IEEE Trans. Microwave Theory Tech., vol. 51, pp. 1–10, Jan. 2003.