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<mathStyle name="User 7" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Math Text Font" font-family="Times New Roman" font-charset="0" font-size="14" font-weight="normal" font-style="normal" underline="false"/> </mathStyles> <dimensionNames mass="mass" length="length" time="time" current="current" thermodynamic-temperature="temperature" luminous-intensity="luminosity" amount-of-substance="substance"/> <symbolics derivation-steps-style="vertical-insert" show-comments="false" evaluate-in-place="false"/> <results> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" exponential-threshold="3" imaginary-value="i" zero-threshold="11"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </results> </mathRendering> <pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="1" orientation="portrait" print-single-page-width="false" page-width="612" page-height="792"> <margins left="86.4" right="86.4" top="86.4" bottom="86.4"/> <header use-full-page-width="false"/> <footer use-full-page-width="false"/> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="EN" UI="EN"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="false" optimize-expressions="false" exact-boolean="true" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="126" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="32" left="6" top="2.25" width="156.75" height="12" align-x="10.5" align-y="12" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">EE 423<sp count="5"/>Power Systems Analysis</p> </text> </region> <region region-id="34" left="6" top="32.25" width="120" height="12" align-x="29.25" align-y="42" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <i>Symmetrical Components</i> </b> </p> </text> </region> <region region-id="37" left="6" top="74.25" width="38.25" height="12" align-x="18.75" align-y="84" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Define a:</p> </text> </region> <region region-id="38" left="54" top="54" width="54.75" height="33.75" align-x="64.5" align-y="84" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">a</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="j">1</ml:imag> <ml:parens> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="42" left="6" top="116.25" width="73.5" height="12" align-x="24.75" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Sequence Matrix</p> </text> </region> <region region-id="40" left="108" top="94.5" width="71.25" height="57.75" align-x="121.5" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">A</ml:id> <ml:matrix rows="3" cols="3"> <ml:real>1</ml:real> <ml:real>1</ml:real> <ml:real>1</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:real>1</ml:real> <ml:id xml:space="preserve">a</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:matrix> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="44" left="192" top="116.25" width="78.75" height="12" align-x="203.25" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">phase A reference</p> </text> </region> <region region-id="47" left="192" top="176.25" width="78.75" height="12" align-x="203.25" align-y="186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">phase B reference</p> </text> </region> <region region-id="45" left="108" top="160.5" width="69.75" height="57.75" align-x="120" align-y="192" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">B</ml:id> <ml:matrix rows="3" cols="3"> <ml:real>1</ml:real> <ml:real>1</ml:real> <ml:real>1</ml:real> <ml:id xml:space="preserve">a</ml:id> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>1</ml:real> <ml:id xml:space="preserve">a</ml:id> </ml:matrix> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="50" left="192" top="242.25" width="78.75" height="12" align-x="203.25" align-y="252" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">phase C reference</p> </text> </region> <region region-id="51" left="108" top="226.5" width="69.75" height="57.75" align-x="120" align-y="258" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">C</ml:id> <ml:matrix rows="3" cols="3"> <ml:real>1</ml:real> <ml:real>1</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:real>1</ml:real> <ml:id xml:space="preserve">a</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="54" left="6" top="320.25" width="164.25" height="24" align-x="18.75" align-y="330" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Define a balanced three phase current</p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">signals</p> </text> </region> <region region-id="55" left="180" top="286.5" width="105.75" height="93.75" align-x="203.25" align-y="336" 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list-style-type="inherit" tabs="inherit">If we used B phase reference sequence matrix and recompute the sequence components</p> </text> </region> <region region-id="85" left="6" top="489" width="75.75" height="22.5" align-x="39" align-y="504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="012_b">I</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">B</ml:id> <ml:real>-1</ml:real> </ml:apply> <ml:id xml:space="preserve" subscript="abc">I</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="87" left="240" top="514.5" width="110.25" height="45.75" align-x="292.5" align-y="540" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math 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xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="012_b">I</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:complex> <ml:real>1.0718850754425932E-13</ml:real> <ml:imag>1.1213252548714056E-14</ml:imag> </ml:complex> <ml:complex> <ml:real>-499.99999999999977</ml:real> <ml:imag>-866.02540378443871</ml:imag> </ml:complex> <ml:complex> <ml:real>5.6843418860808015E-14</ml:real> <ml:imag>5.6843418860808015E-14</ml:imag> </ml:complex> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="13"/> </region> <region region-id="89" left="126" top="517.5" width="97.5" height="51.75" align-x="167.25" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:vectorize/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve" subscript="012_b">I</ml:id> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:real>1.0777343449239695E-13</ml:real> <ml:real>1000</ml:real> <ml:real>8.0388733884609289E-14</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="14"/> </region> <region region-id="67" left="6" top="572.25" width="417" height="24" align-x="18.75" align-y="582" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Notice the magnitude in the positive sequence is identical and the phase shift of -120 is reflected when we used the B phase reference sequence matrix.</p> </text> </region> <region region-id="93" left="6" top="626.25" width="356.25" height="24" align-x="18.75" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Let us examine the sequence components if the magnitudes of phase A and B are</p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">changed to zero (Currents at the fault location for a phase C to ground fault).</p> </text> </region> <region region-id="119" left="6" top="668.25" width="123.75" height="12" align-x="27" align-y="678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Redefining the phase matrix:</p> </text> </region> <region region-id="120" left="144" top="658.5" width="98.25" height="93.75" align-x="167.25" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="abc">I</ml:id> <ml:matrix rows="3" cols="1"> <ml:apply> <ml:mult style="default"/> <ml:real>0</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="j">1</ml:imag> <ml:parens> <ml:real>0</ml:real> </ml:parens> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>0</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="j">1</ml:imag> <ml:parens> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>1000</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="j">1</ml:imag> <ml:parens> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:apply> </ml:matrix> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="121" left="6" top="764.25" width="344.25" height="12" align-x="19.5" align-y="774" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b> <i>Unless otherwise mentioned use phase A reference sequence components:</i> </b> </p> </text> </region> <region region-id="122" left="12" top="783" width="69.75" height="22.5" align-x="34.5" align-y="798" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="012">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">A</ml:id> <ml:real>-1</ml:real> </ml:apply> <ml:id xml:space="preserve" subscript="abc">I</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="123" left="12" top="814.5" width="126.75" height="45.75" align-x="33.75" align-y="840" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="012">I</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:complex> <ml:real>-166.6666666666666</ml:real> <ml:imag>288.6751345948129</ml:imag> </ml:complex> <ml:complex> <ml:real>333.33333333333337</ml:real> <ml:imag>5.6843418860808015E-14</ml:imag> </ml:complex> <ml:complex> <ml:real>-166.66666666666671</ml:real> <ml:imag>-288.6751345948129</ml:imag> </ml:complex> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="17"/> </region> <region region-id="124" left="150" top="814.5" width="87" height="45.75" align-x="180.75" align-y="840" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:vectorize/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve" subscript="012">I</ml:id> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:real>333.33333333333331</ml:real> <ml:real>333.33333333333337</ml:real> <ml:real>333.33333333333343</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="18"/> </region> <region region-id="125" left="246" top="814.5" width="98.25" height="45.75" align-x="288" align-y="840" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:vectorize/> <ml:apply> <ml:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="012">I</ml:id> </ml:apply> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:matrix rows="3" cols="1"> <ml:real>119.99999999999999</ml:real> <ml:real>9.7706639814559226E-15</ml:real> <ml:real>-120.00000000000001</ml:real> </ml:matrix> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="19"/> </region> <region region-id="126" left="6" top="866.25" width="417" height="24" align-x="18.75" align-y="876" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Notice the sequence magnitudes are identical to each other. 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