106 Kirilenko et al.

transverse axes and their analyses based on TE x, y . and TM x, y . modes reduce to a number of two-dimensional 2D. problems involving E- and H-plane discontinuities in a rectangular waveguide. Electromagnetic interactions among the common junction and the filters are calculated on the basis of standard TE, TM modes by carrying out the appropriate conversion of S-matrices from TE x, y ., TM x, y . to those of standard TE and TM modes. The reduction of a 3D problem to a 2D one is important because it results in a great reduction in computer time and storage w17x.

The Key Elements

The key elements in a waveguide E-plane diplexer are a. the T-junction, b. the double sided step junction, and c. the waveguide bifurcation. The solutions for E- and H-plane T-junctions are based on the equivalent two-dimensional mode matching technique as described in w18x. The numerical algorithm in w18x guarantees fast convergence for not only T-junctions but also for cross-junctions and untruncated 908 bends, when compared with other methods. Fast convergence results from the peculiarities of the matrix system obtained from the set of linear equations required to be solved. Unlike the difference sum terms in conventional mode matching technique w19x, the matrix elements have sum terms in their denominators.

The analyses of double step discontinuities encountered in all types of E-plane diplexers, discussed in this paper, do not require a high degree of accuracy because they are the parts of low quality cavities. There exist several methods for analysis of step discontinuities in a waveguide. They are the modified residue calculus technique w20x, the semi-inversion method w21x and the method of moment based on Gegenbaur polynomial basis w22x. However, we chose the simplest mode matching method for our purpose, keeping in mind Mittra’s rule w20x for choosing the number of modes.

The analysis of a waveguide bifurcation requires the highest accuracy because the longitudinal strip diaphragms bind high quality resonators of the filters. We have developed and used a nonconventional numerical algorithm in order to achieve the required accuracy. At first, the associated diffraction problem is reduced to one involving the first kind of Fredholm integral equation for the el ectric field distribution function at the

waveguide junction aperture. Using the second kind of Chebyshev polynomials as the basis and the test functions, it was possible to obtain a system of linear algebraic equations with the matrix elements as combinations of slowly converging series in combination of Bessel functions and the propagation constants of TEm 0 modes of the narrow and the wide waveguides. Further incorporation of asymptotic correction factors speeds up the convergence procedure so drastically that the method surpasses all the known ones, particularly the conventional mode matching technique, in terms of accuracy and economy of calculation.

IV. DESIGN OF CHANNEL FILTERS

Numerous papers have been published in the literature on the design and analysis of E-plane filters. The list of references date back to 1974 w23x. Good accounts of the subject are available in w24]26x. Therefore, we do not elaborate on this topic but discuss the fine tuning of the E-plane filters, pertinent to avoiding time consuming numerical optimization.

All present day E-plane filters synthesis begins with an initial design based on circuit theory. For example, either Cohn’s w27x or Rhode’s w28x K-in- verter formulas are used to start the synthesis procedure. After determination of the K-inverter values, a suitable electromagnetic modeling based on search algorithm is used to obtain the septum and resonator lengths. However, the designed filter invariably turns out to be lower in center frequency and narrower in bandwidth as revealed by measurement or an overall electromagnetic analysis of the filter. The simple reason for the deviation in center frequency is that the synthesis of individual septum widths and resonator lengths based on K-inverter values does not take into consideration the multimode interaction between adjacent septa and the frequency dependence of the K-inverters realized by the septa. In order to alleviate the drawback of the synthesis method, a new approach is used. Figure 2 shows the flow chart for this filter synthesis scheme. The first step of this new approach is a multimode tuning of cavity lengths. The multimode tuning of cavity lengths is done at the center frequency of the filter only and it does not require the recalculation of generalized S-matrices of the septa. The new section length obtained by this method exactly sets the center frequency of the filter.