## The accuracy of electromagnetic equivalence theorem models of microstrip patch antennas

1991
##### Publisher
Stellenbosch : Stellenbosch University
##### Abstract
ENGLISH ABSTRACT: In this report the application of an equivalence theorem model to a patch in a microstrip medium is considered. The problem originated with the application of the Leontovich boundary condition to an equivalent surface current in a moment method technique used by Mosig and Gardiol [1, 2] for the analysis of microstrip antennas and circuits. Their formulation consists of an equivalent electric source Js which satisfies a boundary condition Et = Zs Js. They thus apply a physical boundary condition to an equivalent source, completely ignoring the magnetic sources in the equivalent model. This approximation is analyzed to assess its accuracy. The Leontovich boundary condition is examined and it is found to be applicable to the physical or true current on a patch. The full equivalence model is then developed and compared to the Leontovich model to find the relationship between the true current flowing on the conductor and the equivalent current. It is found that, although the equivalent electric current is equal to the true current on each of the two sides of the patch, an equivalent magnetic current also exists. The contribution of this current to the total field is then examined for a patch on a single dielectric layer above a ground plane. This also gives an idea of what to expect in the case of a multi layered medium. To do the comparison the fields (also called Green's functions) radiated by an electric and a magnetic dipole on the surface of the dielectric are determined from the boundary conditions. This is done in the spectral domain and the spatial fields are then found by an inverse Fourier transform. The spectral functions are too complicated to have closed form solutions in the near field region and thus the integrals are solved numerically using the Hankel transform. The dipoles are scaled according to a relation between the electric and magnetic currents resulting from the application of the equivalence theorem. It is found that, for frequencies up to 30 GHz, the contribution of the magnetic current can be ignored over the whole region p ≠ 0 with an error of less than 0.5%. (This is for a pure copper patch on a substrate with Er = 2.5 and thickness of 0.04 λ.) At the point p = 0 the magnetic current causes a discontinuity in the electric field normal to the dipole axis and tangential to the surface of the dielectric. In the last chapter of this report a theory is developed assuming that the tangential electric fields on top of the patch are negligible in comparison to those on the bottom. The magnetic current then causes a jump in the field to comply with the condition of zero fields in the conductor. This is also the value of the tangential fields directly above the conductor. The boundary condition is then applied on the dielectric side of the patch while ignoring the magnetic sources completely. The value of the error in ignoring the magnetic sources is thus quantified enabling the accuracy of the approximations to be analyzed for any particular application.
AFRIKAANSE OPSOMMINE: Hierdie verslag ondersoek die toepassing van die ekwivalensie stelling op 'n plak in 'n mikrostrook medium. Die probleem het ontstaan uit die toepassing van die Leontovich randvoorwaarde in 'n moment metode tegniek van Mosig en Gardiol [1, 2] vir die analise van mikrostrook antennes en bane. Hulle formulering bestaan uit 'n ekwivalente elektriese oppervlakstroom wat die Leontovich randwaarde Et = Zs Js vir werklike oppervlakstrome bevredig. Hulle maak dus die benadering dat die ekwivalente strome ook die werklike strome is. Verder ignoreer hulle die bydrae van die magnetiese strome tot die uitgestraalde velde. Die Leontovich voorwaarde is ondersoek en daar is gevind dat dit geldig toegepas kan word op die werklike strome op die plak. Die volledige ekwivalente model is opgestel en met die Leontovich voorwaarde vergelyk om die verband tussen die ekwivalente en werklike strome op die plak te vind. Die ekwivalente elektriese stroom is ook die werklike stroom, maar 'n ekwivalente magnetiese stroom word ook benodig. Die bydrae van hierdie magnetiese stroom is vir 'n plak op 'n enkellaag substraat bepaal, maar dit gee ook 'n goeie idee van wat die effek in 'n multilaag struktuur sal wees. Die Et-velde (ook genoem Green se funksies) wat uitgestraal word deur 'n elektriese en magnetiese dipool op die oppervlak van die diëlektrikum, word gebruik om die bydrae van die magnetiese stroom te bepaal. Hierdie velde word in die spektrale gebied vanaf die randwaardes bepaal en dan met 'n inverse Fourier transform terug getransformeer na die ruimtelike gebied. Vanweë die kompleksiteit van hierdie funksies, kan geslote vorm uitdrukkings vir die integrale nie in die nabyveld verkry word nie, en is hulle numeries met behulp van die Hankel transformasie opgelos. Die dipole is dan geskaleer volgens 'n verband tussen die elektriese en magnetiese strome wat deur die ekwivalensie stelling voorgeskryf word. In die gebied p ≠ 0 is die bydrae van die magnetiese dipool, vir frekwensies onder 30 GHz, kleiner as 0.5% van die totale veld. (Dit is vir 'n suiwer koper plak op 'n substraat met Er = 2.5 en dikte van 0.04λ.) By die punt p = 0 veroorsaak die magnetiese dipool 'n diskontenuiteit in die elektriese veld tangensiaal aan die skeidingsvlak en normaal tot die dipool-as. In die ekwivalente model verseker dit dat die velde nul is binne die geleier. In die laaste hoofstuk word 'n teorie ontwikkel deur die aanname te maak dat die velde bo-op die plak weglaatbaar klein is in vergelyking met die onder die plak. Die tangensiale velde direk bo die plak is dan ook nul sodat daar slegs bronne op die onderste oppervlak van die plak benodig word. Deur die randwaarde in die diëlektrikum toe te pas kan die magnetiese strome geignoreer word sonder om die akkuraatheid baie te beinvloed. Die verslag kwantifiseer dus die akkuraatheid van die benaderings sodat bepaal kan word of die benaderings aan die vereistes van 'n spesifieke toepassing voldoen.
##### Description
Study project (M. Ing) -- University of Stellenbosch, 1991.