Three-Phase Magnetomotive Force Vectors
2020-05-12 12:53
This GeoGebra applet demonstrates the three-phase magnetomotive force vectors in electrical machines. Using geometric constructions and vector rotations, it visually illustrates the synthesis and spatial variation of MMF produced by three-phase alternating current. It incorporates vectors, rotations, trigonometry, and complex numbers to explain rotating magnetic fields.
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.581.0" app="classic" platform="d" id="0a54ccc4-39d6-4bd0-a077-3636e981e8c3" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1382" height="744" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.34370477568740954" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1" size="390" window="66,94,700,522" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1355" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="475" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="70" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1355" height="572"/>
<coordSystem xZero="719.2087535447206" yZero="350.22898491990884" scale="18.70989032579228" yscale="18.709890325792298"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="1"/>
<collapsed val="3"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label="eq1" exp="x^(2) + y^(2) = 4" type="conic"/>
<element type="conic" label="eq1">
<show object="true" label="false" ev="4"/>
<objColor r="160" g="160" b="160" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="1" type="30" typeHidden="1" opacity="102"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="-4.0" A3="0.0" A4="0.0" A5="0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Intersect">
<input a0="xAxis" a1="yAxis"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="-0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="α">
<value val="4.22"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="(2 * π)" absoluteScreenLocation="true" width="200.0" x="30.0" y="50.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="2" type="1" playing="true"/>
</element>
<expression label="B" exp="((2 * sin(α)), 0)" type="point" />
<element type="point" label="B">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.7624120516566506" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="A" a1="B"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.7624120516566506" y="0.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="A"/>
</element>
<expression label="phaseA" exp="phaseA(x) = If[π ≤ x ≤ (5 * π), (2 * sin(x))]" type="function"/>
<element type="function" label="phaseA">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="Rotate">
<input a0="phaseA" a1="π / 2"/>
<output a0="phaseA_1"/>
</command>
<element type="curvecartesian" label="phaseA_1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="C" exp="((2 * sin(α)), α + π)" type="point" />
<element type="point" label="C">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.7624120516566506" y="7.361592653589793" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="B" a1="C"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="false" ev="4"/>
<objColor r="32" g="32" b="32" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="7.361592653589793" z="0.0"/>
<lineStyle thickness="1" type="10" typeHidden="1" opacity="191"/>
<coordStyle style="cartesian"/>
<startPoint exp="B"/>
</element>
<expression label="D" exp="((2 * sin(α + (2 / 3 * π))), 0)" type="point" />
<element type="point" label="D">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.06240945760667834" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="D" a1="(2 / 3 * π)"/>
<output a0="D'"/>
</command>
<element type="point" label="D'">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.031204728803339157" y="0.05404817572379142" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="A" a1="D'"/>
<output a0="w"/>
</command>
<element type="vector" label="w">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.031204728803339157" y="0.05404817572379142" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="A"/>
</element>
<command name="Rotate">
<input a0="phaseA" a1="π / 2 + (2 / 3 * π)"/>
<output a0="phaseA_2"/>
</command>
<element type="curvecartesian" label="phaseA_2">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="E" exp="((2 * sin(α + (2 / 3 * π))), α + π + (2 / 3 * π))" type="point" />
<element type="point" label="E">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.06240945760667834" y="9.455987755982989" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="E" a1="(2 / 3 * π)"/>
<output a0="E'"/>
</command>
<element type="point" label="E'">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-8.220330343359214" y="-4.673945702267701" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="D'" a1="E'"/>
<output a0="a"/>
</command>
<element type="vector" label="a">
<show object="true" label="false" ev="4"/>
<objColor r="32" g="32" b="32" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-8.189125614555875" y="-4.7279938779914925" z="0.0"/>
<lineStyle thickness="1" type="10" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="D'"/>
</element>
<expression label="F" exp="((2 * sin(α + (4 / 3 * π))), 0)" type="point" />
<element type="point" label="F">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.700002594049973" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="F" a1="(4 / 3 * π)"/>
<output a0="F'"/>
</command>
<element type="point" label="F'">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.8500012970249873" y="-1.4722454329467207" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="A" a1="F'"/>
<output a0="b"/>
</command>
<element type="vector" label="b">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.8500012970249873" y="-1.4722454329467207" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="A"/>
</element>
<command name="Rotate">
<input a0="phaseA" a1="π / 2 + (4 / 3 * π)"/>
<output a0="phaseA_3"/>
</command>
<element type="curvecartesian" label="phaseA_3">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="G" exp="((2 * sin(α + (4 / 3 * π))), α + π + (4 / 3 * π))" type="point" />
<element type="point" label="G">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.700002594049973" y="11.550382858376183" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="G" a1="(4 / 3 * π)"/>
<output a0="G'"/>
</command>
<element type="point" label="G'">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.152923681765104" y="-7.247436862134817" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="F'" a1="G'"/>
<output a0="c"/>
</command>
<element type="vector" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="32" g="32" b="32" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="10.002924978790091" y="-5.775191429188096" z="0.0"/>
<lineStyle thickness="1" type="10" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="F'"/>
</element>
<expression label="d" exp="u + w + b" type="vector" />
<element type="vector" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="102" g="0" b="153" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.643618077484977" y="-1.4181972572229293" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
</element>
<expression label="eq2" exp="x^(2) + y^(2) = 9" type="conic"/>
<element type="conic" label="eq2">
<show object="true" label="false" ev="4"/>
<objColor r="102" g="0" b="153" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="102"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="-9.0" A3="0.0" A4="0.0" A5="0.0"/>
<eqnStyle style="specific"/>
</element>
</construction>
</geogebra>