*Corresponding author, e-mail: tokan@thk.edu.tr Research Article GU J Sci 31(4): 1156-1164 (2018) Gazi University Journal of Science http://dergipark.gov.tr/gujs Scattering Analysis of Antenna by Using Ludwig Based Hybrid Method Tayfun OKAN 1,* , Nursel AKCAM 2 1 University of Turkish Aeronautical Association, Electrical and Electronics Engineering Department, Ankara, Turkey 2 Gazi University, Electrical-Electronics Engineering Department, Ankara, Turkey Article Info Abstract Solving an electromagnetic problem can be handled in two phases. These are modelling the setup and carrying out the numeric evaluations. Throughout this study, the structure is modelled by Bézier surfaces and the antenna used is meshed with triangular patches. For the calculation part, the method of moments and physical optics (MoM-PO) hybrid method is implemented. While the calculations related with antenna are actualized by using MoM equations, the ones related with structure are obtained by using PO equations. Modified Ludwig’s Algorithm is applied to calculate the current integral for the PO-region. This gives the ability to obtain successful results when the antenna is both close and far from the structure. Overall the stated modelling and calculation technique gives accurate results and saves time and memory in comparison with MoM. Received: 21/03/2018 Accepted: 11/06/2018 Keywords B-spline surfaces method of moments Physical optics Scattering problems Ludwig’s algorithm 1. INTRODUCTION Over the years many studies and techniques have been performed to analyze antennas: exact solution techniques, high frequency asymptotic integration methods and sometimes hybrid methods that combine both. Hybrid approaches can be categorized as either current-based or ray-based. In [1, 2] method of moments-physical optics (MoM-PO) hybrid approach makes the combination of unknown MoM currents and surfaces currents obtained from physical optics (PO). Another application of hybrid MoM-PO method is [3], where MoM is used to solve for surface currents in the electric-field integral equation (EFIE) region, PO is applied for the calculation in the magnetic-field integral equation (MFIE) region. PO is an efficient asymptotic method which calculates the scattered field by approximating the PO current. In order to calculate the oscillatory PO integral, different methods were employed over the past decades. The algorithms that are developed by Gordon [4] and Ludwig [5] are the most basic ones, where Ludwig approximated the phase and amplitude in the integrand by a linear form. Some other special calculation algorithms have been developed, which include the Filon method [6], Levin method [7], asymptotic expansion method [8] and numerical steepest descent path (NSDP) method [9]. Among them Levin method shows good results for complicated phase functions, but it tends to suffer from ill-conditioning. In [9] NSDP method is used to calculate PO integral on the parabolic patch for both monostatic and bistatic RCS calculations. Recently, stationary phase method (SPM) is another widely used technique [10-13] since its computation time is not dependent on frequency. To calculate the PO integral it only takes the effects of some critical points. On the other hand, SPM gives inaccurate results for near fields and additional modifications need to be performed for cases when two critical points are close to each other [10]. Compared to SPM, Ludwig’s Algorithm [14-16] is less efficient but more stable, especially for structures including convex and concave parts.