Določanje električnega polja osno simetričnih struktur

Dolžan Lukežič, Jaka (author), Humar, Iztok (mentor)

V diplomski nalogi je obravnavan problem določanja elektrostatičnega polja v primeru osno simetričnih prevodnih teles. Za takšne strukture ni mogoče vedno uporabiti analitičnih rešitev, zato je potreben numerični pristop, ki omogoča obravnavo splošnih geometrij in večtelesnih konfiguracij. Razvit in implementiran je bil matrični postopek, utemeljen na osnovni Greenovi funkciji $G_{0}$ in lokalnem približku $G_{\mathrm{apr}}$, kjer se kontura telesa razdeli na enakomerno dolge ločne odseke. Sestavljen sistem linearnih enačb omogoča izračun površinskih nabojev, na podlagi katerih so določeni potencial, električno polje in ploskovna gostota naboja. Posebej je bil obravnavan problem singularnosti v lastnih členih matrike $\mathbf{A}_{ii}$, ki je rešen z izolacijo singularnega dela in uporabo Cauchyjeve glavne vrednosti. Metoda je bila preizkušena na preprostih telesih (krogla, elipsoid, cev) ter na kompleksnejši vrtenini, kjer se je pokazala numerična stabilnost in ponovljivost. Rezultati so prikazani s kvantitativnimi grafi in kvalitativnimi vizualizacijami, kot so ekvipotencialne ploskve, silnice polja, barvne karte jakosti polja in porazdelitve naboja. Ugotovljeno je bilo, da se naboji koncentrirajo na območjih ostrih prehodov geometrije, gostota naboja $\sigma$ pa doseže največje vrednosti tam, kjer je jakost polja največja. Metoda je omejena na osno simetrične vrtenine brez navpičnih sten, deluje pa le za idealne prevodnike v homogenem elektrostatičnem okolju. Kot možne nadgradnje se ponujajo adaptivna segmentacija, obravnava različnih dielektričnih medijev ter uporaba Galerkinove metode. Za preverbo je bila metoda primerjana tudi z metodo zrcaljenja na primeru dveh krogel. Analiza je pokazala, da metoda zrcaljenja hitreje konvergira in za praktične potrebe zadošča že z majhnim številom pomožnih nabojev, vendar je omejena na preproste geometrije. Razvita metoda pa se je izkazala kot bolj univerzalno orodje za analizo širšega nabora osno simetričnih struktur.

• Language: Slovenian
• Keywords: elektrostatika, osno simetrična telesa, Greenove funkcije, singularnost, segmentacija, metoda zrcaljenja
• Work type: Bachelor thesis/paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FE - Faculty of Electrical Engineering
• Year: 2025
• PID: 20.500.12556/RUL-173071
• COBISS.SI-ID: 248962051
• Publication date in RUL: 12.09.2025

Secondary language

• Language: English
• Title: Determination of the electric field of axially-symmetric structures

Abstract

The thesis addresses the problem of determining the electrostatic field in the case of axially-symmetric conductive bodies. For such structures analytical solutions are not always applicable, and therefore a numerical approach is required that allows handling general geometries and multi-body configurations. A matrix-based method was developed and implemented, rooted in the fundamental Green’s function $G_{0}$ and a local approximation $G_{\mathrm{apr}}$, where the body’s contour is discretised into arc segments of equal length. The assembled system of linear equations enables calculation of surface charges, on the basis of which the potential, electric field, and surface charge density are determined. In particular, the singularity in the diagonal entries $\mathbf{A}_{ii}$ of the matrix was addressed by isolating the singular part and using the Cauchy principal value. The method was tested on simple bodies (sphere, ellipsoid, tube) as well as on a more complex surface of revolution, where it demonstrated numerical stability and reproducibility. Results are presented through quantitative graphs and qualitative visualisations, such as equipotential surfaces, field lines, colour maps of field magnitude, and charge distributions. It was found that charges concentrate in regions of sharp geometrical transitions, and that the surface charge density $\sigma$ attains its highest values where the field strength is greatest. The method is limited to axially-symmetric bodies of revolution without vertical walls, and applies only to ideal conductors in a homogeneous electrostatic environment. Possible extensions include adaptive segmentation, treatment of different dielectric media, and the use of the Galerkin method. For validation the method was also compared with the method of images in the case of two spheres. The analysis showed that the method of images converges more rapidly and for practical purposes a small number of auxiliary charges is sufficient; however, it is restricted to simple geometries. The developed method has proven to be a more universal tool for analysing a broader class of axially-symmetric structures.

• Keywords: electrostatics, axially-symmetric bodies, Green's functions, singularity, segmentation, method of images