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lock-width="true"> Example </text> </region> <region region-id="1851" left="12" top="36.75" width="57.75" height="14.25" align-x="24.75" align-y="48" 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"> Define Units: </text> </region> <region region-id="1856" left="84" top="36.75" width="60.75" height="15.75" align-x="116.25" align-y="48" 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">kVA</ml:id> <ml:id xml:space="preserve">kW</ml:id> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="1857" left="150" top="36.75" width="69" height="15.75" align-x="190.5" align-y="48" 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">kVAR</ml:id> <ml:id xml:space="preserve">kW</ml:id> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="1905" left="12" top="66" width="392.25" height="43.5" align-x="15" align-y="78" 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"> <f size="12"> <inlineAttr line-through="false">A 3-</inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">f</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> load draws 200kW at a PF of 0.707 lagging from a 440V line. In parallel is a 3-</inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">f</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> capacitor bank that supplies 50kVAR. 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is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Or in one step: </text> </region> <region region-id="616" left="72" top="300" width="135" height="21" align-x="114.75" align-y="312" 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="Load">Q</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="Load">P</ml:id> <ml:apply> <ml:id xml:space="preserve">tan</ml:id> <ml:id xml:space="preserve" subscript="PF">φ</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="555" left="240" top="300.75" width="117" height="20.25" align-x="281.25" align-y="312" 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="Load">Q</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">kVAR</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>200.06040912315493</ml:real> </result> </ml:eval> </math> <rendering item-idref="15"/> </region> <region region-id="1882" left="42" top="336.75" width="367.5" height="14.25" align-x="62.25" align-y="348" 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"> The total reactive power drawn form the source will be </text> </region> <region region-id="1860" left="72" top="360.75" width="136.5" height="20.25" align-x="122.25" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> 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xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="Load">S</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="Load">P</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:parens> <ml:id xml:space="preserve" subscript="Source">Q</ml:id> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="1213" left="258" top="438.75" width="150" height="20.25" align-x="297" align-y="450" 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="Load">S</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">kVA</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:complex> <ml:real>200</ml:real> <ml:imag>150.06040912315493</ml:imag> </ml:complex> </result> </ml:eval> </math> <rendering item-idref="19"/> </region> <region region-id="622" left="72" top="474.75" width="177.75" height="14.25" align-x="79.5" align-y="486" 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"> The resultant power factor angle will be </text> </region> <region region-id="1214" left="258" top="474" width="116.25" height="21" align-x="306.75" align-y="486" 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="PFnew">φ</ml:id> <ml:apply> <ml:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="Load">S</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="1875" left="36" top="510" width="108" height="21" align-x="83.25" align-y="522" show-border="true" show-highlight="true" 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="PFnew">φ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>36.880971841111041</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="21"/> </region> <region region-id="1876" left="156" top="510.75" width="36" height="15.75" align-x="173.25" align-y="522" show-border="true" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">Lagging</ml:id> </math> <rendering item-idref="22"/> </region> <region region-id="1877" left="234" top="510" width="118.5" height="21" align-x="275.25" align-y="522" 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="new">PF</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve" subscript="PFnew">φ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="1217" left="372" top="510.75" width="71.25" height="20.25" align-x="411.75" align-y="522" show-border="true" show-highlight="true" 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="new">PF</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.79988401635602691</ml:real> </result> </ml:eval> </math> <rendering item-idref="24"/> </region> <region region-id="1878" left="30" top="564.75" width="178.5" height="28.5" align-x="37.5" align-y="576" 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"> <f family="Arial" charset="0"> <inlineAttr line-through="false">The current magnitude into the </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false">parallel combination will be </inlineAttr> </f> </text> </region> <region region-id="645" left="228" top="548.25" width="91.5" height="50.25" align-x="259.5" align-y="576" 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="new">I</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:conjugate/> <ml:id xml:space="preserve" subscript="Load">S</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:conjugate/> <ml:id xml:space="preserve" subscript="Line">V</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="1881" left="36" top="612.75" width="160.5" height="20.25" align-x="66" align-y="624" 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="new">I</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:complex> <ml:real>262.431940540739</ml:real> <ml:imag>-196.90322182263384</ml:imag> </ml:complex> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="26"/> </region> <region region-id="1880" left="204" top="612.75" width="106.5" height="20.25" align-x="244.5" align-y="624" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="new">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>328.0874916544538</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="27"/> </region> <region region-id="589" left="330" top="612.75" width="123" height="20.25" align-x="385.5" align-y="624" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="new">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>-36.880971841111041</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="28"/> </region> <region region-id="1883" left="24" top="660.75" width="183" height="28.5" align-x="24" align-y="672" 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"> <f family="Times New Roman" charset="0"> <inlineAttr line-through="false">The current magnitude into the load </inlineAttr> </f> <f family="Times New Roman" charset="0"> <inlineAttr line-through="false">without the parallel capacitor bank is</inlineAttr> </f> </text> </region> <region region-id="1884" left="264" top="666.75" width="157.5" height="20.25" align-x="324.75" align-y="678" 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="Load_old">S</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="Load">P</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:id xml:space="preserve" subscript="Load">Q</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="646" left="66" top="716.25" width="123" height="50.25" align-x="123.75" align-y="744" 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="Load_old">I</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:conjugate/> <ml:id xml:space="preserve" subscript="Load_old">S</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:conjugate/> <ml:id xml:space="preserve" subscript="Line">V</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="653" left="258" top="732.75" width="186.75" height="20.25" align-x="314.25" align-y="744" 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="Load_old">I</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:complex> <ml:real>262.431940540739</ml:real> <ml:imag>-262.51120695781856</ml:imag> </ml:complex> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="31"/> </region> <region region-id="648" left="66" top="774.75" width="132.75" height="20.25" align-x="132.75" align-y="786" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="Load_old">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>371.19086356540168</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="32"/> </region> <region region-id="654" left="258" top="774.75" width="149.25" height="20.25" align-x="339.75" align-y="786" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="Load_old">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>-45.008651662837977</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="33"/> </region> <region region-id="1683" left="30" top="804.75" width="414.75" height="229.5" align-x="39.75" align-y="816" 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"> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false">Note: Comparing the magnitudes of the currents and power factors in both the cases (ie with and without capacitor bank connected in parallel with the load), we see that the load current is higher in the case without capacitor bank and this current is supplied by the source. Since the transmission lines connecting the source and load will have impedance, larger load currents will result in higher losses and larger voltage drops along the line and will ultimately result in poor voltage regulation. So if a capacitor is<sp count="2"/>connected, in parallel with the load, it will meet some of<sp count="2"/>reactive power required by the load and there by reduce the magnitude of the total current (or reactive power)<sp count="2"/>that should be supplied by the source and hence the voltage drop in the line. Thus it provides better voltage regulation in heavily loaded lines. </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false">The same can be explained in terms of power factor. Loads operating at lower power factors will draw large reactive powers. So connecting a capacitor bank in </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false">parallel with the load will<sp count="2"/>support some of the reactive power and there by improve the overall load ( overall load is parallel combination of load and capacitor) power factor and</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false"> reduce the total load current.</inlineAttr> </f> </text> </region> <region region-id="660" left="6" top="1134.75" width="63" height="14.25" align-x="21.75" align-y="1146" 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"> <f family="Times New Roman" charset="0"> <inlineAttr line-through="false">Problem 2.8:</inlineAttr> </f> </text> </region> <region region-id="1885" left="114" top="1134.75" width="246.75" height="69" align-x="117.75" align-y="1146" 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"> In the system shown in Figure P2.8, find I<f family="Arial" charset="0"> <inlineAttr line-through="false"> a </inlineAttr> </f>, I<f family="Arial" charset="0"> <inlineAttr line-through="false"> b </inlineAttr> </f> and I<f family="Arial" charset="0"> <inlineAttr line-through="false"> c </inlineAttr> </f> if (a) Z<f family="Arial" charset="0"> <inlineAttr line-through="false"> a </inlineAttr> </f> = Z<f family="Arial" charset="0"> <inlineAttr line-through="false"> b </inlineAttr> </f> = j1.0, Z<f family="Arial" charset="0"> <inlineAttr line-through="false"> c </inlineAttr> </f> = j 0.9 (b) Za = Zb = Zc = j 1.0 </text> </region> <region region-id="1341" left="72" top="1218.75" width="74.25" height="20.25" align-x="93" align-y="1230" 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="a">Z</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>1.0</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="1342" left="156" top="1218.75" width="44.25" height="20.25" align-x="177.75" align-y="1230" 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="b">Z</ml:id> <ml:id xml:space="preserve" subscript="a">Z</ml:id> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="1343" left="228" top="1218.75" width="74.25" height="20.25" align-x="249" align-y="1230" 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="c">Z</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.9</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="1886" left="12" top="1224.75" width="41.25" height="14.25" align-x="19.5" align-y="1236" 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"> <f family="Times New Roman" charset="0"> <inlineAttr line-through="false">Part (a):</inlineAttr> </f> </text> </region> <region region-id="1887" left="54" top="1249.5" width="86.25" height="21" align-x="77.25" align-y="1266" 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">Va</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0</ml:real> </ml:apply> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">V</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="1890" left="162" top="1249.5" width="103.5" height="21" align-x="185.25" align-y="1266" 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">Vb</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">-1</ml:imag> <ml:real>120</ml:real> </ml:apply> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">V</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="1891" left="288" top="1249.5" width="94.5" height="21" align-x="310.5" align-y="1266" 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">Vc</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>120</ml:real> </ml:apply> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">V</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="1347" left="72" top="1302.75" width="352.5" height="28.5" align-x="82.5" align-y="1314" 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"> Since the system is not a balanced system we cannot apply per phase analysis. Applying KVL along different loops we have: </text> </region> <region region-id="1350" left="72" top="1344.75" width="207" height="18" align-x="263.25" align-y="1356" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Va</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="a">Z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="n">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="40"/> </region> <region region-id="1686" left="72" top="1374.75" width="209.25" height="18" align-x="265.5" align-y="1386" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Vb</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:id xml:space="preserve" subscript="b">Z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="n">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="41"/> </region> <region region-id="1688" left="72" top="1404.75" width="206.25" height="18" align-x="262.5" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Vc</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:id xml:space="preserve" subscript="c">Z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="n">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="42"/> </region> <region region-id="1363" left="72" top="1446.75" width="54.75" height="14.25" align-x="96" align-y="1458" 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"> Substituting </text> </region> <region region-id="1368" left="138" top="1446.75" width="78.75" height="18" align-x="198" align-y="1458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="n">I</ml:id> </ml:apply> </math> <rendering item-idref="43"/> </region> <region region-id="1369" left="222" top="1446.75" width="37.5" height="14.25" align-x="225" align-y="1458" 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"> we have </text> </region> <region region-id="1689" left="78" top="1482.75" width="261" height="18" align-x="323.25" align-y="1494" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Va</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="a">Z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="44"/> </region> <region region-id="1690" left="78" top="1518.75" width="263.25" height="18" align-x="325.5" align-y="1530" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Vb</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:id xml:space="preserve" subscript="b">Z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="45"/> </region> <region region-id="1692" left="78" top="1554.75" width="260.25" height="18" align-x="322.5" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Vc</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:id xml:space="preserve" subscript="c">Z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="46"/> </region> <region region-id="1695" left="60" top="1584.75" width="147" height="14.25" align-x="79.5" align-y="1596" 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"> Collecting terms we are left with: </text> </region> <region region-id="1696" left="84" top="1614.75" width="258.75" height="18" align-x="327" align-y="1626" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Va</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve" subscript="a">Z</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="47"/> </region> <region region-id="1697" left="84" top="1650.75" width="259.5" height="18" align-x="327.75" align-y="1662" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Vb</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve" subscript="b">Z</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="48"/> </region> <region region-id="1698" left="84" top="1686.75" width="258" height="18" align-x="326.25" align-y="1698" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Vc</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve" subscript="c">Z</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="49"/> </region> <region region-id="1700" left="36" top="1722.75" width="281.25" height="14.25" align-x="47.25" align-y="1734" 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"> Three equations and three unknowns. Put this into matrix form: </text> </region> <region region-id="1379" left="48" top="1758.75" width="334.5" height="64.5" align-x="342" align-y="1794" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="default"/> <ml:matrix rows="3" cols="3"> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="a">Z</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="b">Z</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">Z</ml:id> </ml:apply> </ml:matrix> <ml:matrix rows="3" cols="1"> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:matrix> </ml:apply> <ml:matrix rows="3" cols="1"> <ml:id xml:space="preserve">Va</ml:id> <ml:id xml:space="preserve">Vb</ml:id> <ml:id xml:space="preserve">Vc</ml:id> </ml:matrix> </ml:apply> </math> <rendering item-idref="50"/> </region> <region region-id="1480" left="36" top="1845.75" width="332.25" height="76.5" align-x="71.25" align-y="1890" 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:matrix rows="3" cols="1"> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:matrix> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:matrix rows="3" cols="3"> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="a">Z</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="b">Z</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">Z</ml:id> </ml:apply> </ml:matrix> <ml:real>-1</ml:real> </ml:apply> <ml:matrix rows="3" cols="1"> <ml:id xml:space="preserve">Va</ml:id> <ml:id xml:space="preserve">Vb</ml:id> <ml:id xml:space="preserve">Vc</ml:id> </ml:matrix> </ml:apply> </ml:define> </math> <rendering item-idref="51"/> </region> <region region-id="1481" left="222" top="1962.75" width="81" height="20.25" align-x="249" align-y="1974" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="a">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.91233172244913674</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="52"/> </region> <region region-id="1893" left="330" top="1962.75" width="109.5" height="20.25" align-x="372" align-y="1974" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="a">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>-90.350664177004475</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="53"/> </region> <region region-id="1892" left="12" top="1959" width="192" height="71.25" align-x="45.75" align-y="1998" 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:matrix rows="3" cols="1"> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:matrix> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:matrix rows="3" cols="1"> <ml:complex> <ml:real>-0.0055836583093774247</ml:real> <ml:imag>-0.912314635718891</ml:imag> </ml:complex> <ml:complex> <ml:real>-0.79287947993159447</ml:real> <ml:imag>0.45132172791747244</ml:imag> </ml:complex> <ml:complex> <ml:real>0.85988337964412354</ml:real> <ml:imag>0.49645390070921958</ml:imag> </ml:complex> </ml:matrix> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="54"/> </region> <region region-id="1483" left="222" top="1986.75" width="81.75" height="20.25" align-x="249.75" align-y="1998" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.91233172244913685</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="55"/> </region> <region region-id="1894" left="330" top="1986.75" width="109.5" height="20.25" align-x="372.75" align-y="1998" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>150.35066417700449</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="56"/> </region> <region region-id="1485" left="222" top="2010.75" width="81" height="20.25" align-x="249" align-y="2022" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.99290780141843971</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="57"/> </region> <region region-id="1895" left="330" top="2010.75" width="81.75" height="20.25" align-x="372" align-y="2022" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>29.999999999999986</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="58"/> </region> <region region-id="1702" left="36" top="2040.75" width="304.5" height="14.25" align-x="48.75" align-y="2052" 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"> Notice that the currents are still fairly close to balanced three phase. </text> </region> <region region-id="1704" left="60" top="2064.75" width="83.25" height="20.25" align-x="78" align-y="2076" 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="n">I</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="59"/> </region> <region region-id="1706" left="168" top="2064.75" width="81" height="20.25" align-x="195" align-y="2076" 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:absval/> <ml:id xml:space="preserve" subscript="n">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.070921985815602551</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="60"/> </region> <region region-id="1707" left="270" top="2064.75" width="81.75" height="20.25" align-x="312" align-y="2076" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="n">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>29.999999999999712</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="61"/> </region> <region region-id="1710" left="36" top="2094.75" width="99.75" height="14.25" align-x="51" align-y="2106" 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"> Voltage from N' to N: </text> </region> <region region-id="1711" left="48" top="2118.75" width="106.5" height="20.25" align-x="88.5" align-y="2130" 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">Vnp_n</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="n">I</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="62"/> </region> <region region-id="1712" left="174" top="2113.5" width="140.25" height="21" align-x="223.5" align-y="2130" 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:absval/> <ml:id xml:space="preserve">Vnp_n</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.0070921985815602558</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="volt"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="63"/> </region> <region region-id="1896" left="180" top="2148.75" width="153.75" height="14.25" align-x="196.5" align-y="2160" 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"> Roughly 0.71% of source voltage. </text> </region> <region region-id="1490" left="18" top="2184.75" width="45.75" height="14.25" align-x="18" align-y="2196" 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"> <f family="Arial" charset="0"> <inlineAttr line-through="false">Part (b):</inlineAttr> </f> </text> </region> <region region-id="1897" left="72" top="2184.75" width="86.25" height="20.25" align-x="105" align-y="2196" 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="a_2">Z</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>1.0</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="64"/> </region> <region region-id="1580" left="186" top="2184.75" width="87" height="20.25" align-x="219.75" align-y="2196" 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="b_2">Z</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>1.0</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="65"/> </region> <region region-id="1579" left="288" top="2184.75" width="86.25" height="20.25" align-x="321" align-y="2196" 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="c_2">Z</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>1.0</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="66"/> </region> <region region-id="1571" left="60" top="2220.75" width="165.75" height="14.25" align-x="60" align-y="2232" 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"> Applying per phase analysis we have </text> </region> <region region-id="1898" left="42" top="2248.5" width="123.75" height="38.25" align-x="72" align-y="2268" 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="a_2">I</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">Va</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="a_2">Z</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="67"/> </region> <region region-id="1575" left="186" top="2256.75" width="93" height="20.25" align-x="225" align-y="2268" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="a_2">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.90909090909090906</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="68"/> </region> <region region-id="1578" left="288" top="2256.75" width="100.5" height="20.25" align-x="342" align-y="2268" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="a_2">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>-90</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="69"/> </region> <region region-id="1899" left="42" top="2293.5" width="117.75" height="25.5" align-x="72.75" align-y="2310" 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="b_2">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="a_2">I</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">-1</ml:imag> <ml:real>120</ml:real> </ml:apply> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="70"/> </region> <region region-id="1582" left="186" top="2298.75" width="93.75" height="20.25" align-x="225.75" align-y="2310" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="b_2">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.90909090909090906</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="71"/> </region> <region region-id="1583" left="288" top="2298.75" width="100.5" height="20.25" align-x="342.75" align-y="2310" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="b_2">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>150</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="72"/> </region> <region region-id="1900" left="42" top="2335.5" width="108.75" height="25.5" align-x="72" align-y="2352" 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="c_2">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="a_2">I</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>120</ml:real> </ml:apply> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="73"/> </region> <region region-id="1585" left="186" top="2340.75" width="93" height="20.25" align-x="225" align-y="2352" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="c_2">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.90909090909090906</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="74"/> </region> <region region-id="1586" left="288" top="2340.75" width="93.75" height="20.25" align-x="342" align-y="2352" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="c_2">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>29.999999999999986</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> 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</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="1905" 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="1849" left="174" top="12" width="54" height="15" align-x="192" align-y="24" 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"> Example </text> </region> <region region-id="1851" left="12" top="36.75" width="57.75" height="14.25" align-x="24.75" align-y="48" 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"> Define Units: </text> </region> <region region-id="1856" left="84" top="36.75" width="60.75" height="15.75" align-x="116.25" align-y="48" 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">kVA</ml:id> <ml:id xml:space="preserve">kW</ml:id> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="1857" left="150" top="36.75" width="69" height="15.75" align-x="190.5" align-y="48" 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">kVAR</ml:id> <ml:id xml:space="preserve">kW</ml:id> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="1905" left="12" top="66" width="392.25" height="43.5" align-x="15" align-y="78" 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"> <f size="12"> <inlineAttr line-through="false">A 3-</inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">f</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> load draws 200kW at a PF of 0.707 lagging from a 440V line. In parallel is a 3-</inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">f</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> capacitor bank that supplies 50kVAR. Find the resultant power factor and current (magnitude) into the parallel combination.</inlineAttr> </f> </text> </region> <region region-id="1865" left="30" top="138.75" width="87" height="20.25" align-x="70.5" align-y="150" 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="Load">P</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>200</ml:real> <ml:id xml:space="preserve">kW</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="1868" left="150" top="138.75" width="60" height="15.75" align-x="172.5" align-y="150" 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">PF</ml:id> <ml:real>0.707</ml:real> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="1869" left="228" top="138" width="98.25" height="21" align-x="258" align-y="150" 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="PF">φ</ml:id> <ml:apply> <ml:id xml:space="preserve">acos</ml:id> <ml:real>0.707</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="1871" left="348" top="138" width="89.25" height="21" align-x="376.5" align-y="150" 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="PF">φ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>45.008651662837984</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="7"/> </region> <region region-id="1866" left="30" top="168.75" width="73.5" height="20.25" align-x="66.75" align-y="180" 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="Line">V</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>440</ml:real> <ml:id xml:space="preserve">V</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="1870" left="228" top="168.75" width="91.5" height="20.25" align-x="265.5" align-y="180" 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="Cap">Q</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>50</ml:real> <ml:id xml:space="preserve">kVAR</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="1867" left="30" top="192" width="81" height="38.25" align-x="66" align-y="216" 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">Sload</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="Load">P</ml:id> <ml:id xml:space="preserve">PF</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="1678" left="180" top="204.75" width="108" height="15.75" align-x="214.5" align-y="216" 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">Sload</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">kVA</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>282.88543140028287</ml:real> </result> </ml:eval> </math> <rendering item-idref="11"/> </region> <region region-id="1679" left="72" top="246" width="124.5" height="21" align-x="110.25" 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 xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Qload</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">Sload</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve" subscript="PF">φ</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="1872" left="210" top="246.75" width="112.5" height="15.75" align-x="246.75" 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:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Qload</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">kVAR</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>200.06040912315496</ml:real> </result> </ml:eval> </math> <rendering item-idref="13"/> </region> <region region-id="1682" left="48" top="282.75" width="66.75" height="14.25" align-x="52.5" align-y="294" 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"> Or in one step: </text> </region> <region region-id="616" left="72" top="300" width="135" height="21" align-x="114.75" align-y="312" 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="Load">Q</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="Load">P</ml:id> <ml:apply> <ml:id xml:space="preserve">tan</ml:id> <ml:id xml:space="preserve" subscript="PF">φ</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="555" left="240" top="300.75" width="117" height="20.25" align-x="281.25" align-y="312" 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="Load">Q</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">kVAR</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>200.06040912315493</ml:real> </result> </ml:eval> </math> <rendering item-idref="15"/> </region> <region region-id="1882" left="42" top="336.75" width="367.5" height="14.25" align-x="62.25" align-y="348" 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"> The total reactive power drawn form the source will be </text> </region> <region region-id="1860" left="72" top="360.75" width="136.5" height="20.25" align-x="122.25" align-y="372" 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="Source">Q</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="Load">Q</ml:id> <ml:id xml:space="preserve" subscript="Cap">Q</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="1859" left="240" top="366.75" width="124.5" height="20.25" align-x="288.75" align-y="378" 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="Source">Q</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">kVAR</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>150.06040912315493</ml:real> </result> </ml:eval> </math> <rendering item-idref="17"/> </region> <region region-id="619" left="72" top="372.75" width="3" height="14.25" align-x="81.75" align-y="384" 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"> <sp/> </text> </region> <region region-id="620" left="72" top="390.75" width="343.5" height="28.5" align-x="79.5" align-y="402" 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"> Assuming the active power drawn by the load remains constant the apparent power drawn by the load will be </text> </region> <region region-id="621" left="72" top="438.75" width="155.25" height="20.25" align-x="112.5" align-y="450" 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="Load">S</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="Load">P</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:parens> <ml:id xml:space="preserve" subscript="Source">Q</ml:id> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="1213" left="258" top="438.75" width="150" height="20.25" align-x="297" align-y="450" 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="Load">S</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">kVA</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:complex> <ml:real>200</ml:real> <ml:imag>150.06040912315493</ml:imag> </ml:complex> </result> </ml:eval> </math> <rendering item-idref="19"/> </region> <region region-id="622" left="72" top="474.75" width="177.75" height="14.25" align-x="79.5" align-y="486" 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"> The resultant power factor angle will be </text> </region> <region region-id="1214" left="258" top="474" width="116.25" height="21" align-x="306.75" align-y="486" 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="PFnew">φ</ml:id> <ml:apply> <ml:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="Load">S</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="1875" left="36" top="510" width="108" height="21" align-x="83.25" align-y="522" show-border="true" show-highlight="true" 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="PFnew">φ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>36.880971841111041</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="21"/> </region> <region region-id="1876" left="156" top="510.75" width="36" height="15.75" align-x="173.25" align-y="522" show-border="true" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">Lagging</ml:id> </math> <rendering item-idref="22"/> </region> <region region-id="1877" left="234" top="510" width="118.5" height="21" align-x="275.25" align-y="522" 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="new">PF</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve" subscript="PFnew">φ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="1217" left="372" top="510.75" width="71.25" height="20.25" align-x="411.75" align-y="522" show-border="true" show-highlight="true" 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="new">PF</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.79988401635602691</ml:real> </result> </ml:eval> </math> <rendering item-idref="24"/> </region> <region region-id="1878" left="30" top="564.75" width="178.5" height="28.5" align-x="37.5" align-y="576" 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"> <f family="Arial" charset="0"> <inlineAttr line-through="false">The current magnitude into the </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false">parallel combination will be </inlineAttr> </f> </text> </region> <region region-id="645" left="228" top="548.25" width="91.5" height="50.25" align-x="259.5" align-y="576" 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="new">I</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:conjugate/> <ml:id xml:space="preserve" subscript="Load">S</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:conjugate/> <ml:id xml:space="preserve" subscript="Line">V</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="1881" left="36" top="612.75" width="160.5" height="20.25" align-x="66" align-y="624" 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="new">I</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:complex> <ml:real>262.431940540739</ml:real> <ml:imag>-196.90322182263384</ml:imag> </ml:complex> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="26"/> </region> <region region-id="1880" left="204" top="612.75" width="106.5" height="20.25" align-x="244.5" align-y="624" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="new">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>328.0874916544538</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="27"/> </region> <region region-id="589" left="330" top="612.75" width="123" height="20.25" align-x="385.5" align-y="624" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="new">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>-36.880971841111041</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="28"/> </region> <region region-id="1883" left="24" top="660.75" width="183" height="28.5" align-x="24" align-y="672" 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"> <f family="Times New Roman" charset="0"> <inlineAttr line-through="false">The current magnitude into the load </inlineAttr> </f> <f family="Times New Roman" charset="0"> <inlineAttr line-through="false">without the parallel capacitor bank is</inlineAttr> </f> </text> </region> <region region-id="1884" left="264" top="666.75" width="157.5" height="20.25" align-x="324.75" align-y="678" 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="Load_old">S</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="Load">P</ml:id> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:id xml:space="preserve" subscript="Load">Q</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="646" left="66" top="716.25" width="123" height="50.25" align-x="123.75" align-y="744" 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="Load_old">I</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:conjugate/> <ml:id xml:space="preserve" subscript="Load_old">S</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:conjugate/> <ml:id xml:space="preserve" subscript="Line">V</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="653" left="258" top="732.75" width="186.75" height="20.25" align-x="314.25" align-y="744" 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="Load_old">I</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:complex> <ml:real>262.431940540739</ml:real> <ml:imag>-262.51120695781856</ml:imag> </ml:complex> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="31"/> </region> <region region-id="648" left="66" top="774.75" width="132.75" height="20.25" align-x="132.75" align-y="786" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="Load_old">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>371.19086356540168</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="32"/> </region> <region region-id="654" left="258" top="774.75" width="149.25" height="20.25" align-x="339.75" align-y="786" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="Load_old">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>-45.008651662837977</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="33"/> </region> <region region-id="1683" left="30" top="804.75" width="414.75" height="229.5" align-x="39.75" align-y="816" 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"> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false">Note: Comparing the magnitudes of the currents and power factors in both the cases (ie with and without capacitor bank connected in parallel with the load), we see that the load current is higher in the case without capacitor bank and this current is supplied by the source. Since the transmission lines connecting the source and load will have impedance, larger load currents will result in higher losses and larger voltage drops along the line and will ultimately result in poor voltage regulation. So if a capacitor is<sp count="2"/>connected, in parallel with the load, it will meet some of<sp count="2"/>reactive power required by the load and there by reduce the magnitude of the total current (or reactive power)<sp count="2"/>that should be supplied by the source and hence the voltage drop in the line. Thus it provides better voltage regulation in heavily loaded lines. </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false">The same can be explained in terms of power factor. Loads operating at lower power factors will draw large reactive powers. So connecting a capacitor bank in </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false">parallel with the load will<sp count="2"/>support some of the reactive power and there by improve the overall load ( overall load is parallel combination of load and capacitor) power factor and</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false"> reduce the total load current.</inlineAttr> </f> </text> </region> <region region-id="660" left="6" top="1134.75" width="63" height="14.25" align-x="21.75" align-y="1146" 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"> <f family="Times New Roman" charset="0"> <inlineAttr line-through="false">Problem 2.8:</inlineAttr> </f> </text> </region> <region region-id="1885" left="114" top="1134.75" width="246.75" height="69" align-x="117.75" align-y="1146" 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"> In the system shown in Figure P2.8, find I<f family="Arial" charset="0"> <inlineAttr line-through="false"> a </inlineAttr> </f>, I<f family="Arial" charset="0"> <inlineAttr line-through="false"> b </inlineAttr> </f> and I<f family="Arial" charset="0"> <inlineAttr line-through="false"> c </inlineAttr> </f> if (a) Z<f family="Arial" charset="0"> <inlineAttr line-through="false"> a </inlineAttr> </f> = Z<f family="Arial" charset="0"> <inlineAttr line-through="false"> b </inlineAttr> </f> = j1.0, Z<f family="Arial" charset="0"> <inlineAttr line-through="false"> c </inlineAttr> </f> = j 0.9 (b) Za = Zb = Zc = j 1.0 </text> </region> <region region-id="1341" left="72" top="1218.75" width="74.25" height="20.25" align-x="93" align-y="1230" 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="a">Z</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>1.0</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="1342" left="156" top="1218.75" width="44.25" height="20.25" align-x="177.75" align-y="1230" 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="b">Z</ml:id> <ml:id xml:space="preserve" subscript="a">Z</ml:id> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="1343" left="228" top="1218.75" width="74.25" height="20.25" align-x="249" align-y="1230" 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="c">Z</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.9</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="1886" left="12" top="1224.75" width="41.25" height="14.25" align-x="19.5" align-y="1236" 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"> <f family="Times New Roman" charset="0"> <inlineAttr line-through="false">Part (a):</inlineAttr> </f> </text> </region> <region region-id="1887" left="54" top="1249.5" width="86.25" height="21" align-x="77.25" align-y="1266" 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">Va</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0</ml:real> </ml:apply> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">V</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="1890" left="162" top="1249.5" width="103.5" height="21" align-x="185.25" align-y="1266" 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">Vb</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">-1</ml:imag> <ml:real>120</ml:real> </ml:apply> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">V</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="1891" left="288" top="1249.5" width="94.5" height="21" align-x="310.5" align-y="1266" 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">Vc</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>120</ml:real> </ml:apply> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">V</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="1347" left="72" top="1302.75" width="352.5" height="28.5" align-x="82.5" align-y="1314" 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"> Since the system is not a balanced system we cannot apply per phase analysis. Applying KVL along different loops we have: </text> </region> <region region-id="1350" left="72" top="1344.75" width="207" height="18" align-x="263.25" align-y="1356" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Va</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="a">Z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="n">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="40"/> </region> <region region-id="1686" left="72" top="1374.75" width="209.25" height="18" align-x="265.5" align-y="1386" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Vb</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:id xml:space="preserve" subscript="b">Z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="n">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="41"/> </region> <region region-id="1688" left="72" top="1404.75" width="206.25" height="18" align-x="262.5" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Vc</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:id xml:space="preserve" subscript="c">Z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="n">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="42"/> </region> <region region-id="1363" left="72" top="1446.75" width="54.75" height="14.25" align-x="96" align-y="1458" 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"> Substituting </text> </region> <region region-id="1368" left="138" top="1446.75" width="78.75" height="18" align-x="198" align-y="1458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="n">I</ml:id> </ml:apply> </math> <rendering item-idref="43"/> </region> <region region-id="1369" left="222" top="1446.75" width="37.5" height="14.25" align-x="225" align-y="1458" 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"> we have </text> </region> <region region-id="1689" left="78" top="1482.75" width="261" height="18" align-x="323.25" align-y="1494" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Va</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="a">Z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="44"/> </region> <region region-id="1690" left="78" top="1518.75" width="263.25" height="18" align-x="325.5" align-y="1530" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Vb</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:id xml:space="preserve" subscript="b">Z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="45"/> </region> <region region-id="1692" left="78" top="1554.75" width="260.25" height="18" align-x="322.5" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Vc</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:id xml:space="preserve" subscript="c">Z</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="46"/> </region> <region region-id="1695" left="60" top="1584.75" width="147" height="14.25" align-x="79.5" align-y="1596" 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"> Collecting terms we are left with: </text> </region> <region region-id="1696" left="84" top="1614.75" width="258.75" height="18" align-x="327" align-y="1626" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Va</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve" subscript="a">Z</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id 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xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve" subscript="b">Z</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="48"/> </region> <region region-id="1698" left="84" top="1686.75" width="258" height="18" align-x="326.25" align-y="1698" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Vc</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="c">I</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve" subscript="c">Z</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="49"/> </region> <region region-id="1700" left="36" top="1722.75" width="281.25" height="14.25" align-x="47.25" align-y="1734" 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"> Three equations and three unknowns. Put this into matrix form: </text> </region> <region region-id="1379" left="48" top="1758.75" width="334.5" height="64.5" align-x="342" align-y="1794" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:mult style="default"/> <ml:matrix rows="3" cols="3"> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="a">Z</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> 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<ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">Z</ml:id> </ml:apply> </ml:matrix> <ml:matrix rows="3" cols="1"> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:matrix> </ml:apply> <ml:matrix rows="3" cols="1"> <ml:id xml:space="preserve">Va</ml:id> <ml:id xml:space="preserve">Vb</ml:id> <ml:id xml:space="preserve">Vc</ml:id> </ml:matrix> </ml:apply> </math> 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<ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">Z</ml:id> </ml:apply> </ml:matrix> <ml:real>-1</ml:real> </ml:apply> <ml:matrix rows="3" cols="1"> <ml:id xml:space="preserve">Va</ml:id> <ml:id xml:space="preserve">Vb</ml:id> <ml:id xml:space="preserve">Vc</ml:id> </ml:matrix> </ml:apply> </ml:define> </math> <rendering item-idref="51"/> </region> <region region-id="1481" left="222" top="1962.75" width="81" height="20.25" align-x="249" align-y="1974" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="a">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.91233172244913674</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="52"/> </region> <region region-id="1893" left="330" top="1962.75" width="109.5" height="20.25" align-x="372" align-y="1974" show-border="true" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> 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xml:space="preserve" subscript="b">I</ml:id> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:matrix> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:matrix rows="3" cols="1"> <ml:complex> <ml:real>-0.0055836583093774247</ml:real> <ml:imag>-0.912314635718891</ml:imag> </ml:complex> <ml:complex> <ml:real>-0.79287947993159447</ml:real> <ml:imag>0.45132172791747244</ml:imag> </ml:complex> <ml:complex> <ml:real>0.85988337964412354</ml:real> <ml:imag>0.49645390070921958</ml:imag> </ml:complex> </ml:matrix> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="54"/> </region> <region region-id="1483" left="222" top="1986.75" width="81.75" height="20.25" align-x="249.75" align-y="1998" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.91233172244913685</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="55"/> </region> <region region-id="1894" left="330" top="1986.75" width="109.5" height="20.25" align-x="372.75" align-y="1998" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> 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height="20.25" align-x="372" align-y="2022" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>29.999999999999986</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="58"/> </region> <region region-id="1702" left="36" top="2040.75" width="304.5" height="14.25" align-x="48.75" align-y="2052" 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"> Notice that the currents are still fairly close to balanced three phase. </text> </region> <region region-id="1704" left="60" top="2064.75" width="83.25" height="20.25" align-x="78" align-y="2076" 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="n">I</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="59"/> </region> <region region-id="1706" left="168" top="2064.75" width="81" height="20.25" align-x="195" align-y="2076" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> 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<result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>29.999999999999712</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="61"/> </region> <region region-id="1710" left="36" top="2094.75" width="99.75" height="14.25" align-x="51" align-y="2106" 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"> Voltage from N' to N: </text> </region> <region region-id="1711" left="48" top="2118.75" width="106.5" height="20.25" align-x="88.5" align-y="2130" 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">Vnp_n</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="n">I</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="62"/> </region> <region region-id="1712" left="174" top="2113.5" width="140.25" height="21" align-x="223.5" align-y="2130" 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:absval/> <ml:id xml:space="preserve">Vnp_n</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.0070921985815602558</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="volt"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="63"/> </region> <region region-id="1896" left="180" top="2148.75" width="153.75" height="14.25" align-x="196.5" align-y="2160" 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"> Roughly 0.71% of source voltage. </text> </region> <region region-id="1490" left="18" top="2184.75" width="45.75" height="14.25" align-x="18" align-y="2196" 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"> <f family="Arial" charset="0"> <inlineAttr line-through="false">Part (b):</inlineAttr> </f> </text> </region> <region region-id="1897" left="72" top="2184.75" width="86.25" height="20.25" align-x="105" align-y="2196" 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="a_2">Z</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>1.0</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="64"/> </region> <region region-id="1580" left="186" top="2184.75" width="87" height="20.25" align-x="219.75" align-y="2196" 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="b_2">Z</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>1.0</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="65"/> </region> <region region-id="1579" left="288" top="2184.75" width="86.25" height="20.25" align-x="321" align-y="2196" 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="c_2">Z</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>1.0</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="66"/> </region> <region region-id="1571" left="60" top="2220.75" width="165.75" height="14.25" align-x="60" align-y="2232" 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"> Applying per phase analysis we have </text> </region> <region region-id="1898" left="42" top="2248.5" width="123.75" height="38.25" align-x="72" align-y="2268" 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="a_2">I</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">Va</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="a_2">Z</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="67"/> </region> <region region-id="1575" left="186" top="2256.75" width="93" height="20.25" align-x="225" align-y="2268" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="a_2">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.90909090909090906</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="68"/> </region> <region region-id="1578" left="288" top="2256.75" width="100.5" height="20.25" align-x="342" align-y="2268" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="a_2">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>-90</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="69"/> </region> <region region-id="1899" left="42" top="2293.5" width="117.75" height="25.5" align-x="72.75" align-y="2310" 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="b_2">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="a_2">I</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">-1</ml:imag> <ml:real>120</ml:real> </ml:apply> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="70"/> </region> <region region-id="1582" left="186" top="2298.75" width="93.75" height="20.25" align-x="225.75" align-y="2310" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="b_2">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.90909090909090906</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="71"/> </region> <region region-id="1583" left="288" top="2298.75" width="100.5" height="20.25" align-x="342.75" align-y="2310" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="b_2">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>150</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="72"/> </region> <region region-id="1900" left="42" top="2335.5" width="108.75" height="25.5" align-x="72" align-y="2352" 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="c_2">I</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="a_2">I</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">e</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>120</ml:real> </ml:apply> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="73"/> </region> <region region-id="1585" left="186" top="2340.75" width="93" height="20.25" align-x="225" align-y="2352" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="c_2">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.90909090909090906</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="74"/> </region> <region region-id="1586" left="288" top="2340.75" width="93.75" height="20.25" align-x="342" align-y="2352" show-border="true" show-highlight="true" 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:id xml:space="preserve">arg</ml:id> <ml:id xml:space="preserve" subscript="c_2">I</ml:id> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>29.999999999999986</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="75"/> </region> <region region-id="1716" left="12" top="2376.75" width="245.25" height="14.25" align-x="17.25" align-y="2388" 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"> As a check, use the matrix equations set up for part A: </text> </region> <region region-id="1717" left="30" top="2397.75" width="368.25" height="76.5" align-x="65.25" align-y="2442" 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:matrix rows="3" cols="1"> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:matrix> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:matrix rows="3" cols="3"> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="a_2">Z</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="b_2">Z</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.1</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult style="default"/> <ml:imag font="0" symbol="j">1</ml:imag> <ml:real>0.2</ml:real> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c_2">Z</ml:id> </ml:apply> </ml:matrix> <ml:real>-1</ml:real> </ml:apply> <ml:matrix rows="3" cols="1"> <ml:id xml:space="preserve">Va</ml:id> <ml:id xml:space="preserve">Vb</ml:id> <ml:id xml:space="preserve">Vc</ml:id> </ml:matrix> </ml:apply> </ml:define> </math> <rendering item-idref="76"/> </region> <region region-id="1718" left="216" top="2496.75" width="81" height="20.25" align-x="243" align-y="2508" show-border="true" show-highlight="true" 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:absval/> <ml:id xml:space="preserve" subscript="a">I</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>0.90909090909090917</ml:real> <unitMonomial 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top="2493" width="154.5" height="71.25" align-x="69.75" align-y="2532" 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:matrix rows="3" cols="1"> <ml:id xml:space="preserve" subscript="a">I</ml:id> <ml:id xml:space="preserve" subscript="b">I</ml:id> <ml:id xml:space="preserve" subscript="c">I</ml:id> </ml:matrix> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:matrix rows="3" cols="1"> <ml:complex> <ml:imag>-0.90909090909090917</ml:imag> </ml:complex> <ml:complex> <ml:real>-0.7872958216222169</ml:real> <ml:imag>0.45454545454545436</ml:imag> </ml:complex> <ml:complex> <ml:real>0.7872958216222169</ml:real> <ml:imag>0.45454545454545436</ml:imag> </ml:complex> </ml:matrix> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> 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