{"id":2001,"date":"2022-09-29T12:41:34","date_gmt":"2022-09-29T03:41:34","guid":{"rendered":"https:\/\/www.ssil.co.jp\/product\/EMSolution\/en2\/?post_type=basic_ems&#038;p=2001"},"modified":"2022-10-14T11:50:08","modified_gmt":"2022-10-14T02:50:08","slug":"b04_1","status":"publish","type":"basic_ems","link":"https:\/\/www.ssil.co.jp\/product\/EMSolution\/en\/about\/basic_ems\/b04_1\/","title":{"rendered":"4-1. A-\u03c6 method"},"content":{"rendered":"<div style=\"text-align: justify;\">\n<font color=\"firebrick\"><strong>EMSolution<\/strong><\/font> uses a finite element method with edge elements based on the <span class=\"red\"><strong>$A$-$\\phi$ method<\/strong><\/span> which uses magnetic vector potential $A$ and electric scalar potential $\\phi$.<sup>[17]<\/sup><\/div>\n<div><strong>The $A$-$\\phi$ method<\/strong> is a fundamental formulation usually used in magnetic field analysis, along with the <strong>$T$-$\\Omega$ method<\/strong>, but its characteristics compared to <strong>the $T$-$\\Omega$ method<\/strong> are as follows: <\/div>\n<p>&nbsp;<\/p>\n<ul class=\"list01\">\n<li><strong>Easy handling of eddy current multi-connected conductors\u2019 problems<\/strong><\/li>\n<p>&nbsp;<\/p>\n<li>Good convergence in nonlinear problems (apparently)<\/li>\n<p>&nbsp;<\/p>\n<li>Easy to extend to high-frequency problems including displacement currents (apparently)<\/li>\n<p>&nbsp;\n<\/ul>\n<div style=\"text-align: justify;\">\nOn the other hand, in the $T$-$\\Omega$ method, the non-conductive region can be treated with a scalar function $\\Omega$, whereas the $A$-$\\phi$ method requires the use of a vector function $A$, so\n<\/div>\n<p>&nbsp;<\/p>\n<ul class=\"list01\">\n<li>It has a disadvantage of greater freedom of analysis<\/li>\n<p>&nbsp;\n<\/ul>\n<p>&nbsp;<\/p>\n<div style=\"text-align: justify;\">\nIt is known that <strong>the $A$-$\\phi$ and $T$-$\\Omega$ methods<\/strong> have a <strong>dual relationship<\/strong> and that, for example, in a static magnetic field analysis, the inte-grated magnetic energy is sandwiched between the values obtained by <strong>both methods<\/strong>. From the viewpoint of error evaluation, analysis by both methods may be considered in the future.\n<\/div>\n<p>&nbsp;<\/p>\n<div style=\"text-align: justify;\">\nIn <font color=\"firebrick\"><strong>EMSolution<\/strong><\/font>, the $A$-$\\phi$ method is basically used, but the magnetic scalar potential $\\Omega$ is assumed to be available for the air domain to reduce the number of unknowns. However, the convergence of the ICCG method is quite poor, probably due to the loss of positivity of the system matrix, and it does not help much in reducing the computation time.<sup>[18]<\/sup> It is used only in cases where there is insufficient computer capacity.\n<\/div>\n<p>&nbsp;<\/p>\n<div style=\"text-align: justify;\">\nAnother feature of <font color=\"firebrick\"><strong>EMSolution<\/strong><\/font> is the use of <strong>the reduced magnetic potential $A_r$<\/strong> for the air domain (<span class=\"red\">2-potential method<\/span>), where $A_r$ represents the contribution of the magnetic field due to eddy currents and magnetization in the analysis domain, separated from the source magnetic field. The source magnetic field is obtained from the Biot-Savart law.\n<\/div>\n<p>&nbsp;<\/p>\n<div style=\"text-align: justify;\">\nThe advantage of using Ar is that\n<\/div>\n<ul class=\"list01\">\n<li><strong>Source currents can be represented independently of the analysis mesh<\/strong><\/li>\n<p>&nbsp;<\/p>\n<li>The magnetic field due to the source current is not included, so <strong>there is no need to make the mesh so fine near the source current.<\/strong><\/li>\n<p>&nbsp;<\/p>\n<li><strong>Source current can be defined outside the analysis area<\/strong><\/li>\n<p>&nbsp;<\/p>\n<li><strong>Source currents can move freely within the analysis do-main<\/strong>, facilitating analysis in the presence of moving con-ductors.<sup>[10]<\/sup><\/li>\n<p>&nbsp;\n<\/ul>\n<div style=\"text-align: justify;\">\nThe two-potential method is often said to take a large amount of computation time to calculate the source magnetic field. This method requires integration of the source magnetic vector potential on the edges and the magnetic field strength ($H$) on the plane at the boundary between the total potential region (where $A$ is the variable) and the reduced potential region (where $A_r$ is the variable). These need only be calculated once and are not such a large burden in EMSolution, except when the source is displaced. There are various analytical integrals for integrating the Biot-Savart law, and you will still need to use them.<sup>[19]<\/sup>\n<\/div>\n<div class=\"img col1\">\n<div>\n\t<a href=\"\/product\/EMSolution\/en\/wp-content\/uploads\/image411.jpg\" class=\"modal\"><br \/>\n\t<img decoding=\"async\" src=\"\/product\/EMSolution\/en\/wp-content\/uploads\/image411.jpg\" alt=\"\" \/><\/a>\n  <\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>EMSolution uses a finite element method with edge elements based on the $A$-$\\phi$ method which uses magnetic vector potential $A$ and electric scalar potential $\\phi$.[17] The $A$-$\\phi$ method is a fundamental formulation usually used in magnetic field analysis, along with the $T$-$\\Omega$ method, but its characteristics compared to the $T$-$\\Omega$ method are as follows: &nbsp; [&hellip;]<\/p>\n","protected":false},"featured_media":0,"parent":0,"template":"","class_list":["post-2001","basic_ems","type-basic_ems","status-publish","hentry"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.ssil.co.jp\/product\/EMSolution\/en\/wp-json\/wp\/v2\/basic_ems\/2001"}],"collection":[{"href":"https:\/\/www.ssil.co.jp\/product\/EMSolution\/en\/wp-json\/wp\/v2\/basic_ems"}],"about":[{"href":"https:\/\/www.ssil.co.jp\/product\/EMSolution\/en\/wp-json\/wp\/v2\/types\/basic_ems"}],"wp:attachment":[{"href":"https:\/\/www.ssil.co.jp\/product\/EMSolution\/en\/wp-json\/wp\/v2\/media?parent=2001"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}