Mathcad Professional 14.1 <description/> <author>Electrical and Computer Engr.</author> <company>Parametric Technology Corporation</company> <keywords/> <revisedBy>Patrice</revisedBy> </userData> <identityInfo> <revision>2</revision> <documentID>697B68F2-33CA-495C-BB7D-0AA8BFF7E0A6</documentID> <versionID>BE8D583B-FAFB-4D5B-8468-E29E849F8D5B</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Heading 1"> <blockAttr margin-left="0" margin-right="0" 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display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </results> </mathRendering> <pageModel show-page-frame="false" show-header-frame="true" show-footer-frame="true" header-footer-start-page="1" paper-code="1" orientation="landscape" print-single-page-width="false" page-width="792" page-height="612"> <margins left="86.4" right="86.4" top="86.4" bottom="86.4"/> <header use-full-page-width="false"> <left>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fnil\fprq11\fcharset0 Arial;}} \viewkind4\uc1\pard\f0\fs18 ECE 320\par Energy Systems I\par }</left> <center>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fnil\fprq11\fcharset0 Arial;}} \viewkind4\uc1\pard\qc\f0\fs18 Buck Converter\par Fourier Series of v_x(t)\par }</center> <right>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fnil\fprq11\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\f0\fs18 HW #9\par Solution\par }</right> </header> <footer use-full-page-width="false"> <left>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fnil\fprq11\fcharset0 Arial;}} \viewkind4\uc1\pard\f0\fs18\{p\}\par \{f\}\par }</left> <center>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fnil\fprq11\fcharset0 Arial;}} \viewkind4\uc1\pard\qc\f0\fs18 Page \{n\} of \{nn\}\par }</center> <right>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fnil\fprq11\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\f0\fs18 November 13, 2010\par }</right> </footer> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="en" UI="en"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="false" optimize-expressions="false" exact-boolean="true" strings-use-origin="false" zero-over-zero="error"> <compatibility multiple-assignment="MC12" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="true" 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="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="1707" 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="350" left="6" top="5.25" width="140.25" height="15.75" align-x="8.25" align-y="18" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">I. Define the Situation</p> </text> </region> <region region-id="1267" left="576" top="7.5" width="35.25" height="14.25" align-x="586.5" align-y="18" 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/math30"> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:sqrt/> <ml:real>-1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="372" left="30" top="38.25" width="304.5" height="12" align-x="41.25" 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="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Given the following buck converter, parameters, currents, and voltages.</p> </text> </region> <region region-id="377" left="54" top="66" width="411.75" height="158.25" align-x="54" align-y="66" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{00021A14-0000-0000-C000-000000000046}" type="static" item-idref="3"/> <rendering item-idref="4"/> </region> <region region-id="1109" left="138" top="254.25" width="72.75" height="12" align-x="151.5" align-y="264" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>Source Voltage</b> </p> </text> </region> <region region-id="1108" left="228" top="254.25" width="90" height="24" align-x="247.5" align-y="264" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>Switching Period and Duty Cycle</b> </p> </text> </region> <region region-id="1106" left="360" top="254.25" width="59.25" height="24" align-x="373.5" align-y="264" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>Output Filter</b> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>Parameters</b> </p> </text> </region> <region region-id="1107" left="486" top="254.25" width="69.75" height="12" align-x="499.5" align-y="264" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>Output Current</b> </p> </text> </region> <region region-id="1080" left="30" top="279" width="72" height="18.75" align-x="56.25" align-y="294" 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/math30"> <ml:id xml:space="preserve">μsec</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-6</ml:real> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="1081" left="138" top="285" width="43.5" height="16.5" align-x="155.25" align-y="294" 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/math30"> <ml:id xml:space="preserve" subscript="s">V</ml:id> <ml:apply> <ml:mult/> <ml:real>10</ml:real> <ml:id xml:space="preserve">V</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="1093" left="228" top="285" width="57" height="16.5" align-x="245.25" align-y="294" 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/math30"> <ml:id xml:space="preserve" subscript="s">T</ml:id> <ml:apply> <ml:mult/> <ml:real>10</ml:real> <ml:id xml:space="preserve">μsec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="1083" left="360" top="285" width="49.5" height="12.75" align-x="372" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">L</ml:id> <ml:apply> <ml:mult/> <ml:real>200</ml:real> <ml:id xml:space="preserve">μH</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="1084" left="492" top="285" width="38.25" height="16.5" align-x="507" align-y="294" 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/math30"> <ml:id xml:space="preserve" subscript="o">I</ml:id> <ml:apply> <ml:mult/> <ml:real>7</ml:real> <ml:id xml:space="preserve">A</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="1092" left="228" top="308.25" width="36.75" height="31.5" align-x="243" align-y="324" 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/math30"> <ml:id xml:space="preserve" subscript="s">f</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="1086" left="360" top="315" width="48.75" height="12.75" align-x="372.75" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:mult/> <ml:real>470</ml:real> <ml:id xml:space="preserve">μF</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="1091" left="228" top="351" width="52.5" height="16.5" align-x="241.5" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="s">f</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">kHz</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>100.00000000000001</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="kilohertz"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="12"/> </region> <region region-id="1090" left="228" top="375" width="33" height="12.75" align-x="241.5" align-y="384" 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/math30"> <ml:id xml:space="preserve">D</ml:id> <ml:real>0.8</ml:real> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="343" left="6" top="443.25" width="52.5" height="15.75" align-x="8.25" align-y="456" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">II. Goals</p> </text> </region> <region region-id="1173" left="12" top="470.25" width="444" height="15" align-x="19.5" align-y="480" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">a) Plot v<sub>x</sub>(t) versus time for time equal to<sp count="2"/>0 to T<sub>s</sub> <sub> <sp/> </sub>using MathCAD's programming tools (the vertical line).</p> </text> </region> <region region-id="290" left="12" top="506.25" width="373.5" height="30" align-x="19.5" align-y="516" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="4.5" margin-right="0" text-indent="-4.5" text-align="left" list-style-type="inherit" tabs="inherit">b) Plot v<sub>x</sub>(t) and v<sub>xFS</sub>(t) versus time for time equal to<sp count="2"/>0 to T<sub>s</sub> <sub> <sp/> </sub>on the same graph.</p> <p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="inherit" tabs="13.5"> <tab/>Where v<sub>xFS</sub>(t) is the Fourier Series of<sp count="2"/>v<sub>x</sub>(t). </p> </text> </region> <region region-id="298" left="12" top="554.25" width="494.25" height="57" align-x="19.5" align-y="564" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="inherit" tabs="inherit">c) Plot v<sub>O</sub>(t) and v<sub>OFS</sub>(t) versus time for time equal to<sp count="2"/>0 to T<sub>s</sub> <sub> <sp/> </sub>on the same graph.</p> <p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="inherit" tabs="13.5"> <tab/>Where v<sub>O</sub>(t) is the voltage across the capacitor of a series connection of a capacitor and inductor across v<sub>x</sub>(t) and v<sub>OFS</sub>(t) is the Fourier Series of<sp count="2"/>v<sub>O</sub>(t)obtained from circuit analysis and v<sub>xFS</sub>(t).</p> <p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="inherit" tabs="13.5"> <tab/>Values for the inductor and capacitor are given above.</p> </text> </region> <region region-id="1113" left="6" top="641.25" width="96" height="15.75" align-x="8.25" align-y="654" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">III. Take Action</p> </text> </region> <region region-id="1119" left="30" top="662.25" width="444" height="15" align-x="37.5" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">a) Plot v<sub>x</sub>(t) versus time for time equal to<sp count="2"/>0 to T<sub>s</sub> <sub> <sp/> </sub>using MathCAD's programming tools (the vertical line).</p> </text> </region> <region region-id="1207" left="54" top="735" width="46.5" height="16.5" align-x="72" 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:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math30"> <originRef 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xml:space="preserve" subscript="o">V</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve" subscript="s">V</ml:id> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="14"> <element-image-map> <box left="1.5" top="0.75" width="45" height="15" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="1208" left="198" top="735" width="42" height="16.5" align-x="214.5" align-y="744" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:provenance expr-id="1"> <originRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="248653300" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="359752500" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">a4494d417eb5cc6e429b8bfad7b43fea</contentHash> <ml:id xml:space="preserve" subscript="o">V</ml:id> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>8</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="volt"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="15"> <element-image-map> <box left="1.5" top="0.75" width="10.5" height="15" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="1213" left="414" top="734.25" width="198" height="12" align-x="435.75" align-y="744" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Alternative ways to represent the upper bound</p> </text> </region> <region region-id="1209" left="198" top="760.5" width="168" height="51.75" align-x="224.25" align-y="780" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="x">v</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:ifThen> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="s">T</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:ifThen> <ml:otherwise> <ml:id xml:space="preserve" subscript="s">V</ml:id> </ml:otherwise> </ml:program> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="1210" left="420" top="760.5" width="168" height="31.5" align-x="533.25" align-y="780" 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/math30"> <ml:equal/> <ml:apply> <ml:equal/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="s">T</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">D</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:minus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">D</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="17"/> </region> <region region-id="1182" left="6" top="906" width="590.25" height="381.75" align-x="6" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="18"/> <rendering item-idref="19"/> </region> <region region-id="1215" left="30" top="1328.25" width="373.5" height="30" align-x="37.5" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="4.5" margin-right="0" text-indent="-4.5" text-align="left" list-style-type="inherit" tabs="inherit">b) Plot v<sub>x</sub>(t) and v<sub>xFS</sub>(t) versus time for time equal to<sp count="2"/>0 to T<sub>s</sub> <sub> <sp/> </sub>on the same graph.</p> <p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="inherit" tabs="13.5"> <tab/>Where v<sub>xFS</sub>(t) is the Fourier Series of<sp count="2"/>v<sub>x</sub>(t). </p> </text> </region> <region region-id="1259" left="432" top="1324.5" width="88.5" height="31.5" align-x="441.75" align-y="1344" 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/math30"> <ml:id xml:space="preserve">t</ml:id> <ml:range> <ml:sequence> <ml:apply> <ml:mult/> <ml:real>0</ml:real> <ml:id xml:space="preserve">μsec</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="s">T</ml:id> <ml:real>1000</ml:real> </ml:apply> </ml:sequence> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="1224" left="48" top="1383" width="51" height="16.5" align-x="77.25" align-y="1392" 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/math30"> <ml:id xml:space="preserve" subscript="max">N</ml:id> <ml:real>101</ml:real> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="1223" left="108" top="1383" width="63.75" height="16.5" align-x="120" align-y="1392" 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/math30"> <ml:id xml:space="preserve">n</ml:id> <ml:range> <ml:sequence> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:sequence> <ml:id xml:space="preserve" subscript="max">N</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="1222" left="216" top="1372.5" width="92.25" height="31.5" align-x="231.75" align-y="1392" 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/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve" subscript="s">V</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="1225" left="360" top="1376.25" width="221.25" height="36.75" align-x="396.75" align-y="1392" 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/math30"> <ml:function> <ml:id xml:space="preserve" subscript="xFS">v</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve" subscript="s">V</ml:id> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult/> <ml:apply> <ml:absval/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">arg</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="1228" left="54" top="1422" width="519.75" height="307.5" align-x="54" align-y="1422" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="25"/> <rendering item-idref="26"/> </region> <region region-id="1230" left="18" top="1766.25" width="494.25" height="57" align-x="25.5" align-y="1776" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="inherit" tabs="inherit">c) Plot v<sub>O</sub>(t) and v<sub>OFS</sub>(t) versus time for time equal to<sp count="2"/>0 to T<sub>s</sub> <sub> <sp/> </sub>on the same graph.</p> <p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="inherit" tabs="13.5"> <tab/>Where v<sub>O</sub>(t) is the voltage across the capacitor of a series connection of a capacitor and inductor across v<sub>x</sub>(t) and v<sub>OFS</sub>(t) is the Fourier Series of<sp count="2"/>v<sub>O</sub>(t)obtained from circuit analysis and v<sub>xFS</sub>(t).</p> <p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="inherit" tabs="13.5"> <tab/>Values for the inductor and capacitor are given above.</p> </text> </region> <region region-id="313" left="54" top="1840.5" width="87.75" height="31.5" align-x="72" align-y="1860" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math30"> <originRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="361173548" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> 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subscript="s">T</ml:id> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="27"> <element-image-map> <box left="1.5" top="0.75" width="86.25" height="30" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="314" left="198" top="1851" width="54.75" height="12.75" align-x="214.5" align-y="1860" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:provenance expr-id="1"> <originRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="360996828" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" 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version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="360996468" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">a875f8dc73515a9e1231c344a22ad4bf</contentHash> <ml:define> <ml:id xml:space="preserve" subscript="min">I</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="o">I</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">ΔI</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="29"> <element-image-map> <box left="1.5" top="0.75" width="65.25" height="26.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="316" left="198" top="1887" width="60.75" height="16.5" align-x="220.5" align-y="1896" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:provenance expr-id="1"> <originRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="360996628" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="360996828" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">92974de4388f1bd06e6bd5ac5bbc4c3b</contentHash> <ml:id xml:space="preserve" subscript="min">I</ml:id> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>6.96</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="30"> <element-image-map> <box left="1.5" top="0.75" width="16.5" height="15" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="317" left="54" top="1929" width="73.5" height="16.5" align-x="78.75" align-y="1938" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math30"> <originRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="361173148" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="361173388" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">7bfb44f153b5b38da0ab024e5dddfaa8</contentHash> <ml:define> <ml:id xml:space="preserve" subscript="max">I</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="min">I</ml:id> <ml:id xml:space="preserve">ΔI</ml:id> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="31"> <element-image-map> <box left="1.5" top="0.75" width="72" height="15" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="318" left="198" top="1929" width="61.5" height="16.5" align-x="221.25" align-y="1938" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:provenance expr-id="1"> <originRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="361173388" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="361173148" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">6141fcbef6abb00335a129520c6b2e37</contentHash> <ml:id xml:space="preserve" subscript="max">I</ml:id> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>7.04</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ampere"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="32"> <element-image-map> <box left="1.5" top="0.75" width="17.25" height="15" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="319" left="198" top="1960.5" width="307.5" height="102" align-x="222.75" align-y="1980" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="L">i</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:ifThen> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="s">V</ml:id> <ml:id xml:space="preserve" subscript="o">V</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">t</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve" subscript="min">I</ml:id> </ml:apply> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessThan/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="s">T</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve" subscript="o">V</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve" subscript="max">I</ml:id> </ml:apply> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessThan/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="s">T</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="s">V</ml:id> <ml:id xml:space="preserve" subscript="o">V</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="s">T</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve" subscript="min">I</ml:id> </ml:apply> </ml:ifThen> </ml:program> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="1296" left="54" top="2084.25" width="280.5" height="15" align-x="61.5" align-y="2094" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The Fourier Series of i<sub>L</sub>(t) is also not required, but is done for fun.</p> </text> </region> <region region-id="1316" left="90" top="2108.25" width="51.75" height="31.5" align-x="108" align-y="2124" 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/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="1317" left="186" top="2108.25" width="93.75" height="33" align-x="203.25" align-y="2124" 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/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">C</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="1318" left="294" top="2103" width="219.75" height="42" align-x="329.25" align-y="2124" 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/math30"> <ml:function> <ml:id xml:space="preserve" subscript="LFS">i</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="o">I</ml:id> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult/> <ml:apply> <ml:absval/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">arg</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="1571" left="72" top="2204.25" width="395.25" height="15" align-x="75" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">A graph of i<sub>L</sub>(t) is not required, but I used it to confirm that the expression for i<sub>L</sub>(t) is correct.</p> </text> </region> <region region-id="1563" left="18" top="2226" width="525" height="403.5" align-x="18" align-y="2226" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="37"/> <rendering item-idref="38"/> </region> <region region-id="1400" left="60" top="2667" width="72" height="16.5" align-x="85.5" align-y="2676" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math30"> <originRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="248652820" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="360996828" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">8c4e8c50eb1f1c2e5e9601812c434079</contentHash> <ml:define> <ml:function> <ml:id xml:space="preserve" subscript="C">i</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="L">i</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="o">I</ml:id> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="39"> <element-image-map> <box left="1.5" top="0.75" width="70.5" height="15" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="1418" left="60" top="2701.5" width="138.75" height="41.25" align-x="94.5" align-y="2724" 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/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve" subscript="o">Δv</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve" subscript="C">i</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="atzero">V</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="40"/> </region> <region region-id="1417" left="246" top="2715" width="30" height="16.5" align-x="264" align-y="2724" 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/math30"> <ml:id xml:space="preserve" subscript="o">Δv</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </math> <rendering item-idref="41"/> </region> <region region-id="1416" left="294" top="2714.25" width="120.75" height="12" align-x="300.75" align-y="2724" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">has zero average value; i.e.,</p> </text> </region> <region region-id="1415" left="444" top="2692.5" width="155.25" height="61.5" align-x="587.25" align-y="2724" 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/math30"> <ml:equal/> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve" subscript="C">i</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="atzero">V</ml:id> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:bounds> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="42"/> </region> <region region-id="1346" left="60" top="2752.5" width="137.25" height="61.5" align-x="103.5" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math30"> <originRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="248652980" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="360996828" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">302f247ca458c5fdd31be22d7e4eea5c</contentHash> <ml:define> <ml:id xml:space="preserve">Constant</ml:id> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve" subscript="C">i</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:bounds> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="43"> <element-image-map> <box left="1.5" top="0.75" width="135.75" height="60" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="1345" left="264" top="2768.25" width="90" height="31.5" align-x="300.75" align-y="2784" 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/math30"> <ml:id xml:space="preserve" subscript="atzero">V</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">Constant</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="44"/> </region> <region region-id="1347" left="438" top="2769" width="109.5" height="22.5" align-x="473.25" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="atzero">V</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-8.5102431398817438E-05</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="volt"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="45"/> </region> <region region-id="1348" left="60" top="2821.5" width="141" height="41.25" align-x="96" align-y="2844" 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/math30"> <ml:function> <ml:id xml:space="preserve" subscript="o">Δv</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve" subscript="C">i</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="atzero">V</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="1423" left="306" top="2840.25" width="48" height="12" align-x="309.75" align-y="2850" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">as a check</p> </text> </region> <region region-id="1422" left="396" top="2823.75" width="99.75" height="45" align-x="461.25" align-y="2850" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:provenance expr-id="1"> <originRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="248652820" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="2BFD841A-9587-4DC2-80CB-4BC292E7B61F" version-id="C6BD7076-36DA-45F4-A152-0A953B24DDC8" branch-id="00000000-0000-0000-0000-000000000000" revision-num="248652860" is-modified="false" region-id="0" href="C:\JLAW\CLASSES\2010 Fall 320\HANDOUTS\PowerElectronics\s28 idealbuckR2.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">de3ef0819e4447e26f2194e38f40dd99</contentHash> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve" subscript="o">Δv</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve" subscript="s">T</ml:id> </ml:bounds> </ml:apply> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-1.48230119534649E-25</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="weber"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="47"> <element-image-map> <box left="1.5" top="0.75" width="59.25" height="43.5" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="1693" left="12" top="2912.25" width="282" height="12" align-x="21" align-y="2922" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">ΔVo is not required, but it is presented for educational purposes. </p> </text> </region> 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xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="second"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="50"/> </region> <region region-id="1697" left="372" top="2937" width="93" height="16.5" align-x="412.5" align-y="2946" 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/math30"> <ml:id xml:space="preserve" subscript="omax">Δv</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="o">Δv</ml:id> <ml:id xml:space="preserve" subscript="max">t</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="51"/> </region> <region region-id="1698" left="498" top="2937" width="82.5" height="16.5" align-x="537" align-y="2946" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="omax">Δv</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">mV</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.12766849436203448</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="millivolt"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="52"/> </region> <region region-id="1699" left="12" top="2967" width="44.25" height="16.5" align-x="36" align-y="2976" 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/math30"> <ml:id xml:space="preserve" subscript="min">t</ml:id> <ml:apply> <ml:mult/> <ml:real>0</ml:real> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:define> 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is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="min">t</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="second"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="55"/> </region> <region region-id="1702" left="372" top="2967" width="91.5" height="16.5" align-x="411.75" align-y="2976" 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/math30"> <ml:id xml:space="preserve" subscript="omin">Δv</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="o">Δv</ml:id> <ml:id 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