Investigation of the solution relative convergence when calculating step irregularities in the transmission line by the partial domain method using the energy orthogonality condition

The absolute convergence of diffraction problems on stepwise irregularities in transmission lines using the energy orthogonality condition is beyond doubt. In this article, we consider the question of relative convergence and investigate the dependence of the accuracy of the obtained solution on the order of the reduced system, i.e., on the number of waves taken into account in the connected transmission lines. We believe that the most accurate results for a finite number of waves taken into account in the connected waveguides are obtained when the complex powers of the waves propagating in both waveguides are equal. In this case, for the phase constants the relation is valid: β εμjm j j Sk ωεμ β2 k k  kn2   , β εμkn k k Sj ωεμ β2 j j  jm2  where βjm – the phase constant of a wave with maximum number m considered in a waveguide j; βkn – the phase constant of a wave with maximum number n considered in a waveguide k; Sj, Sk – the cross-sectional areas of waveguides j and k, respectively. Using the result of Weyl's spectral theorem, we obtain: N N S Sjk   j k34 , where Nj, Nk – number of considered waves in waveguides j and k, respectively. When calculating step irregularities, when the areas of the connected waveguides differ significantly from each other, the number of waves taken into account in the field expansions must be taken sufficiently large to ensure equality of the tangential components of the field at the interface. The use of the relations obtained in the article allows you to choose this number as optimal, and to achieve the specified accuracy of the solution with a smaller number of waves taken into account, which leads to a reduction in time and computational resources.