class MultiPhase "Multi phase windings"
extends Modelica.Icons.Information;
annotation (Documentation(info = "<html>\n<h4>Symmetrical three phase system</h4>\n\n<p>\nSymmetrical three phases systems of currents (or voltages) consists of three sinusoidal\nsine waves with an angular displacement of\n<img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/2pi_3.png\"/>.\n</p>\n\n<p>\n<img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/i123.png\"/>,\n</p>\n\n<p>\nElectrical three phase machines have (usually) symmetrical three phase windings which\nexcite spatial magnetic potential with a spacial displacement of\n<img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/2pi_3.png\"/>\n- with respect to the fundamental wave,\nsee [<a href=\"modelica://Modelica.Magnetic.FundamentalWave.UsersGuide.References\">Laughton02</a>].\nSuch a symmetrical three phase system of currents (or voltages) can be represented by\n<a href=\"http://en.wikipedia.org/wiki/Phasor\">phasors</a>, as depicted in Fig. 1(a).\nThe associated three phase winding is depicted in Fig. 2(a). The winding axis are displaced by\n<img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/2pi_3.png\"/>:\n</p>\n\n<p>\n<img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/orientationk_3.png\"/>\n</p>\n\n<p>\nSo there is a strong coherence between angular displacement in the time and\nspatial domain which also applies to multi phase systems.\n</p>\n\n<table border=\"0\" cellspacing=\"0\" cellpadding=\"2\">\n <caption align=\"bottom\"><strong>Fig. 1: </strong>Symmetrical (a) three phase and (b) five phase current system</caption>\n <tr>\n <td>\n <img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/phase35.png\"\n alt=\"phase35.png\">\n </td>\n </tr>\n</table>\n\n<table border=\"0\" cellspacing=\"0\" cellpadding=\"2\">\n <caption align=\"bottom\"><strong>Fig. 2: </strong>Symmetrical (a) three phase and (b) five phase winding</caption>\n <tr>\n <td>\n <img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/winding35.png\"\n alt=\"winding35.png\">\n </td>\n </tr>\n</table>\n\n<h4>Symmetrical multi phase system</h4>\n\n<p>\nIn symmetrical multi phase systems odd and even phase numbers have to be distinguished.\n</p>\n\n<h5>Odd number of phases</h5>\n\n<p>\nFor a symmetrical multi phase system with <img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/m.png\"/>\nphases the displacement in the time and spatial domain is\n<img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/2pi_m.png\"/>,\nas depicted in Fig. 1 and 2.\n</p>\n\n<p>\nMathematically, this symmetry is expressed in terms of phase currents by:\n</p>\n\n<p><img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/ik_odd.png\"/></p>\n\n<p>\nThe orientation of the winding axis of such winding is given by:\n</p>\n\n<p><img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/orientationk_m.png\"/></p>\n\n<h5>Even number of phases</h5>\n\n<p>\nIn the current implementation of the FundamentalWave library, phase numbers equal\nto the power of two are not supported. However, any other multi phase system with even\nan phase number, <img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/m.png\"/>,\ncan be recursively split into various symmetrical systems with odd phase numbers, as depicted in Fig. 3 and 4.\nThe displacement between the two symmetrical systems is\n<img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/pi_m.png\"/>.\nA function for calculating the <a href=\"modelica://Modelica.Electrical.MultiPhase.Functions.symmetricOrientation\">symmetricOrientation</a> is available.\n</p>\n\n<table border=\"0\" cellspacing=\"0\" cellpadding=\"2\">\n <caption align=\"bottom\"><strong>Fig. 3: </strong>Symmetrical (a) six and (b) ten phase current system</caption>\n <tr>\n <td>\n <img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/phase610.png\"\n alt=\"phase610.png\">\n </td>\n </tr>\n</table>\n\n<table border=\"0\" cellspacing=\"0\" cellpadding=\"2\">\n <caption align=\"bottom\"><strong>Fig. 4: </strong>Symmetrical (a) six and (b) ten phase winding</caption>\n <tr>\n <td>\n <img src=\"modelica://Modelica/Resources/Images/Magnetic/FundamentalWave/UsersGuide/MultiPhase/winding610.png\"\n alt=\"winding610.png\">\n </td>\n </tr>\n</table>\n\n\n<h4>Note</h4>\n\n<p>\nIn a fully symmetrical machine, the orientation of the winding axes and the symmetrical currents (or voltages)\n<a href=\"http://en.wikipedia.org/wiki/Phasor\">phasors</a> have different signs; see Fig. 1 and 2 for odd phase numbers\nand Fig. 3 and 4 for even phase numbers.\n</p>\n</html>"));
end MultiPhase;