Abstract— In this research the Expressions that determine the eigenfunctions and modes eigenvalues in waveguides with a composite sectorial cross-section are obtained. The possibility of characteristics changing for eigenvalues by changing the parameters that characterizing the cross-sectional shape was studied. The field modes of waveguide based on Ritz method was analyzed.it was determined the characteristics of the quasi –   modes in a cruciform sector waveguide, and quasi –   modes in a composite sector waveguide with an arbitrary number of sectors.it was also shown the advantage of using the cross-section waveguide in single mode optical fiber wavelength range. The eigenvalues () and the normalized coefficients ( )for quasi –   modes in terms of Bessel functions (  ,  ) and their combinations was obtained. Keywords—Signal Processing, Signals, Communications, Signal Processing Systems, Communication Systems, Waveguides, cross section, eigenfunctions, eigenvalues, mode. I. INTRODUCTION In communication and signal processing technology, as well as in several other electrical devices, the using of metal and dielectric waveguides is enough widespread. In this case, most often waveguides have rectangular or circular cross- sectional shape. The task of determining the eigenfunctions and eigenvalues of modes in such waveguides is relatively simple to solve since the lines that bounding the cross-sectional contour coincide with the coordinate surfaces of the rectangular and cylindrical coordinate systems, which does not cause difficulties in imposing boundary conditions for determining the integration constants when solving the wave equation [1, 2]. In this case, as a rule, the variable separation method is used [1]. A special case of guiding systems in particular, waveguides having a complex cross-sectional shape, where the boundaries partially coincide with the coordinate surfaces of the selected coordinate system, for example as a cruciform L-shaped, H-shaped and others. In this case, to solve the wave equation, it is possible to apply approximate solution methods, such as the method of partial domains [2], the method of associated equations [3], and others. In addition to the above waveguide cross-sectional shapes, waveguides that having a composite sectorial cross-sectional shape are practically interesting to research, a special case of such shape is a waveguide with a cross sectorial sectional shape (fig. 1). The advantage of such a cross-sectional shape is that their electromagnetic waves as studies have shown [4], preserve the structure of the waveguides wave field with a circular cross- sectional shape and, at the same time, such a shape allows changing the characteristics of the (eigenvalues) within specified limits by changing the parameters characterizing the cross-sectional shape [5,6,7]. (a) (b) Fig. 1: waveguide with a cross sectorial sectional shape The dielectric waveguide which having the shape shown in (Fig. 1b) can be used in the optical wavelength range [8,9]. In other words, if the core of the optical fiber is given a shape like this figure, then in a single-mode operation of optical fiber the coupling between ordinary and extraordinary waves will significantly decrease in comparison with a fiber with a circular cross-section, and the crosstalk between the waves will increase [4,10,11]. As a result, the polarization mode Mohammed Yousef AL-Gawagzeh. Department of Electrical power Engineering, Faculty of Engineering Al-Balqa Applied University, Jordan Received: November 27. Revised: December 29, 2020. Accepted: December 29, 2020. Determination the Modes Characteristics in the Complex Cross-Section Waveguides INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2020.14.138 Volume 14, 2020 ISSN: 1998-4464 1103