<description/> <author>Electrical and Computer Engr.</author> <company>Parametric Technology Corporation</company> <keywords/> <revisedBy>Electrical and Computer Engr.</revisedBy> </userData> <identityInfo> <revision>7</revision> <documentID>B97AE31D-CA64-4A68-8EFC-B430C8E979FA</documentID> <versionID>9F13DB68-70BB-4EC4-99D0-5C1F76AAFC87</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" text-indent="inherit" 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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="portrait" print-single-page-width="false" page-width="612" page-height="792"> <margins left="86.4" right="86.4" top="86.4" bottom="86.4"/> <header use-full-page-width="false"> <left>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fnil\fprq11\fcharset0 Arial;}} \viewkind4\uc1\pard\f0\fs18 ECE 520: AEM\par }</left> <center>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fnil\fprq11\fcharset0 Arial;}} \viewkind4\uc1\pard\qc\f0\fs18 S30\par }</center> <right>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fnil\fprq11\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\f0\fs18 Equal Area Criteria\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\}\{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 April 28, 2008\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="223" 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="14" left="6" top="6.75" width="125.25" height="13.5" align-x="18.75" 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"> I. Define the Situation: </text> </region> <region region-id="9" left="48" top="27" width="28.5" height="12.75" align-x="64.5" align-y="36" 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">pu</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="4" left="48" top="51" width="48" height="12.75" align-x="60.75" align-y="60" 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">H</ml:id> <ml:apply> <ml:mult/> <ml:real>2.5</ml:real> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="6" left="48" top="68.25" width="76.5" height="27.75" align-x="71.25" align-y="84" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="eB">ω</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>60</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">rad</ml:id> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="7" left="168" top="68.25" width="78" height="27.75" align-x="190.5" align-y="84" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="eB">ω</ml:id> <ml:unitOverride> <ml:apply> <ml:div/> <ml:id xml:space="preserve">rad</ml:id> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>376.99111843077515</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> <unitReference unit="second" power-numerator="-1"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="5"/> </region> <region region-id="8" left="48" top="105" width="59.25" height="16.5" align-x="75" align-y="114" 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">T</ml:id> <ml:apply> <ml:mult/> <ml:real>2.1</ml:real> <ml:id xml:space="preserve">pu</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="12" left="18" top="146.25" width="416.25" height="72" align-x="21" align-y="156" 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"> A synchronous generator with the parameters given above is operating with constant mechanical torque, Tm, equal to 0.7 pu and delivering power to an infinite bus.<sp count="2"/>The electrical power (torque) generated by the generator goes to zero for a period of time, tc, (clearing time).<sp count="2"/>The clearing angle, <f family="Symbol" charset="2">d</f> c, is the angle of the rotor at tc.<sp count="2"/>The critical clearing time, tcr, is the maximum clearing time for stable operation.<sp count="2"/>The critical clearing angle, <f family="Symbol" charset="2">d</f> cr, is the angle of the rotor at tcr. </text> </region> <region region-id="32" left="24" top="231" width="51" height="16.5" align-x="42.75" align-y="240" 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="m">T</ml:id> <ml:apply> <ml:mult/> <ml:real>0.7</ml:real> <ml:id xml:space="preserve">pu</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="16" left="18" top="266.25" width="58.5" height="12" align-x="33.75" align-y="276" 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"> Assume that: </text> </region> <region region-id="20" left="54" top="297" width="152.25" height="57" align-x="89.25" align-y="306" 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="e">T</ml:id> <ml:sequence> <ml:id xml:space="preserve">δ</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> <ml:program> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="max">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">t</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:greaterOrEqual/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve" subscript="c">t</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="max">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:ifThen> </ml:program> </ml:apply> </math> <rendering item-idref="8"/> </region> <region region-id="22" left="6" top="372.75" width="57" height="13.5" align-x="9.75" 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="false"> II. Desired </text> </region> <region region-id="24" left="36" top="410.25" width="23.25" height="12" align-x="43.5" align-y="420" 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"> Part I </text> </region> <region region-id="26" left="54" top="440.25" width="284.25" height="72" align-x="65.25" align-y="450" 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"> Using analytical methods (not simulation) calcuate the: <tab/>1. critical clearing angle, <f family="Symbol" charset="2">d</f> cr, <tab/>2. critical clearing time, tcr, <tab/>3. the maximum swing angle, <f family="Symbol" charset="2">d</f> maxswing (with tc = 0.8 tcr), and <tab/>4. the minimum swing angle, minswing (with tc = 0.8 tcr). </text> </region> <region region-id="27" left="12" top="630.75" width="65.25" height="13.5" align-x="15.75" align-y="642" 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"> III. Solution </text> </region> <region region-id="36" left="36" top="656.25" width="183" height="15" align-x="37.5" align-y="666" 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"> 1. Calculate the critical clearing angle, δcr. </text> </region> <region region-id="39" left="54" top="686.25" width="161.25" height="15" align-x="57" align-y="696" 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"> a. Calculate the initial rotor angle, δ0. </text> </region> <region region-id="40" left="72" top="711" width="99" height="16.5" align-x="111.75" align-y="720" 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:id xml:space="preserve" subscript="e">T</ml:id> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="max">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve" subscript="0">δ</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="9"/> </region> <region region-id="110" left="72" top="742.5" width="75" height="35.25" align-x="87" align-y="762" 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="0">δ</ml:id> <ml:apply> <ml:id xml:space="preserve">asin</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="m">T</ml:id> <ml:id xml:space="preserve" subscript="max">T</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="108" left="246" top="753" width="52.5" height="16.5" align-x="260.25" align-y="762" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="0">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.33983690945412193</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="11"/> </region> <region region-id="109" left="348" top="753" width="63.75" height="16.5" align-x="362.25" align-y="762" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="0">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>19.471220634490692</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="12"/> </region> <region region-id="47" left="54" top="788.25" width="78" height="15" align-x="57" align-y="798" 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"> b. Calculate δmax. </text> </region> <region region-id="51" left="144" top="788.25" width="219.75" height="12" align-x="151.5" align-y="798" 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"> The following is only valid for round rotor machines. </text> </region> <region region-id="56" left="78" top="819" width="60" height="16.5" align-x="103.5" align-y="828" 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">δ</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve" subscript="0">δ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="55" left="240" top="819" width="67.5" height="16.5" align-x="264.75" align-y="828" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="max">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.8017557441356713</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="14"/> </region> <region region-id="59" left="348" top="819" width="78.75" height="16.5" align-x="372.75" align-y="828" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="max">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>160.52877936550931</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="15"/> </region> <region region-id="63" left="54" top="854.25" width="72" height="15" align-x="55.5" align-y="864" 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"> c. Calculate δ.cr. </text> </region> <region region-id="70" left="78" top="884.25" width="54.75" height="12" align-x="88.5" align-y="894" 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"> Initial guess: </text> </region> <region region-id="87" left="162" top="874.5" width="65.25" height="31.5" align-x="176.25" align-y="894" 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="x">δ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="max">δ</ml:id> <ml:id xml:space="preserve" subscript="0">δ</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="86" left="258" top="885" width="56.25" height="16.5" align-x="271.5" align-y="894" 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="x">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.5707963267948966</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="17"/> </region> <region region-id="88" left="354" top="885" width="47.25" height="16.5" align-x="367.5" align-y="894" 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="x">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <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="18"/> </region> <region region-id="71" left="72" top="921" width="24" height="12.75" align-x="83.25" align-y="930" show-border="false" show-highlight="false" 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/math30">Given</ml:id> </math> <rendering item-idref="19"/> </region> <region region-id="77" left="102" top="939.75" width="174.75" height="48.75" align-x="150" align-y="966" 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" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">δ</ml:id> </ml:boundVars> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve" subscript="0">δ</ml:id> <ml:id xml:space="preserve" subscript="x">δ</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">δ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="max">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve" subscript="x">δ</ml:id> <ml:id xml:space="preserve" subscript="max">δ</ml:id> </ml:bounds> </ml:apply> </ml:apply> </math> <rendering item-idref="20"/> </region> <region region-id="93" left="72" top="1017" width="59.25" height="16.5" align-x="90" align-y="1026" 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="cr">δ</ml:id> <ml:apply> <ml:Find auto-method="true" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:id xml:space="preserve" subscript="x">δ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="92" left="222" top="1017" width="60" height="16.5" align-x="239.25" align-y="1026" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.69327172026567</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="91" left="330" top="1017" width="66.75" height="16.5" align-x="347.25" align-y="1026" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>97.017323140079426</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="23"/> </region> <region region-id="96" left="48" top="1064.25" width="159" height="12" align-x="49.5" align-y="1074" 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"> 2. Calcuate the critical clearing time: </text> </region> <region region-id="100" left="84" top="1084.5" width="102" height="37.5" align-x="101.25" align-y="1110" 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:id xml:space="preserve" subscript="cr">δ</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="eB">ω</ml:id> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">H</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve" subscript="cr">t</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="0">δ</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="24"/> </region> <region region-id="104" left="240" top="1100.25" width="109.5" height="12" align-x="249.75" align-y="1110" 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"> From Session 20 page 9. </text> </region> <region region-id="105" left="84" top="1134.75" width="109.5" height="33" align-x="100.5" align-y="1152" 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="cr">t</ml:id> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">H</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="eB">ω</ml:id> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="cr">δ</ml:id> <ml:id xml:space="preserve" subscript="0">δ</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="107" left="240" top="1149" width="59.25" height="16.5" align-x="255.75" align-y="1158" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">t</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.22646656448963631</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="second"/> </unitMonomial> </unitedValue> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="4" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="26"/> </region> <region region-id="112" left="48" top="1262.25" width="238.5" height="15" align-x="49.5" align-y="1272" 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"> 3. Calculate the maximum swing angle with tc = 0.8 tcr. </text> </region> <region region-id="113" left="72" top="1299" width="45.75" height="16.5" align-x="85.5" align-y="1308" 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="c">t</ml:id> <ml:apply> <ml:mult/> <ml:real>0.8</ml:real> <ml:id xml:space="preserve" subscript="cr">t</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="114" left="252" top="1299" width="56.25" height="16.5" align-x="264.75" align-y="1308" 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="c">t</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.18117325159170905</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="second"/> </unitMonomial> </unitedValue> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="4" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="28"/> </region> <region region-id="119" left="72" top="1336.5" width="97.5" height="37.5" align-x="87" align-y="1362" 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="c">δ</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="eB">ω</ml:id> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">H</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve" subscript="c">t</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="0">δ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="120" left="252" top="1353" width="57" height="16.5" align-x="266.25" align-y="1362" 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="c">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.2060351883735128</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="30"/> </region> <region region-id="153" left="342" top="1353" width="63.75" height="16.5" align-x="356.25" align-y="1362" 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="c">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>69.1007262380675</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="31"/> </region> <region region-id="146" left="72" top="1406.25" width="54.75" height="12" align-x="82.5" align-y="1416" 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"> Initial guess: </text> </region> <region region-id="147" left="156" top="1396.5" width="68.25" height="31.5" align-x="170.25" 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:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="x">δ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="max">δ</ml:id> <ml:id xml:space="preserve" subscript="cr">δ</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="150" left="252" top="1407" width="56.25" height="16.5" align-x="265.5" 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:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="x">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.2475137322006704</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> 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background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="35"/> </region> <region region-id="177" left="102" top="1467.75" width="164.25" height="48.75" align-x="150" 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/math30"> <ml:equal/> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">δ</ml:id> </ml:boundVars> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve" subscript="0">δ</ml:id> <ml:id xml:space="preserve" subscript="c">δ</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">δ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="max">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve" subscript="c">δ</ml:id> <ml:id xml:space="preserve" subscript="x">δ</ml:id> </ml:bounds> </ml:apply> </ml:apply> </math> <rendering item-idref="36"/> </region> <region region-id="178" left="72" top="1545" width="90.75" height="16.5" align-x="121.5" align-y="1554" 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="maxswing">δ</ml:id> <ml:apply> <ml:Find auto-method="true" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:id xml:space="preserve" subscript="x">δ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="179" left="216" top="1545" width="91.5" height="16.5" align-x="264.75" align-y="1554" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="maxswing">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.6513172729909318</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="38"/> </region> <region region-id="180" left="324" top="1545" width="98.25" height="16.5" align-x="372.75" align-y="1554" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="maxswing">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>94.6135103794328</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="39"/> </region> <region region-id="200" left="66" top="1598.25" width="238.5" height="15" align-x="67.5" align-y="1608" 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"> 3. Calculate the maximum swing angle with tc = 0.8 tcr. </text> </region> <region region-id="215" left="84" top="1670.25" width="54.75" height="12" align-x="94.5" align-y="1680" 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"> Initial guess: </text> </region> <region region-id="216" left="168" top="1664.25" width="36.75" height="27.75" align-x="182.25" align-y="1680" 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/math30"> <ml:id xml:space="preserve" subscript="x">δ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>4</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="40"/> </region> <region region-id="217" left="264" top="1671" width="61.5" height="16.5" align-x="277.5" align-y="1680" 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="x">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-0.78539816339744828</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="41"/> </region> <region region-id="218" left="354" top="1671" width="52.5" height="16.5" align-x="367.5" align-y="1680" 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="x">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-45</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="42"/> </region> <region region-id="219" left="84" top="1713" width="24" height="12.75" align-x="95.25" align-y="1722" show-border="false" show-highlight="false" 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/math30">Given</ml:id> </math> <rendering item-idref="43"/> </region> <region region-id="220" left="114" top="1731.75" width="155.25" height="48.75" align-x="258" align-y="1758" 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" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">δ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="max">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve" subscript="maxswing">δ</ml:id> <ml:id xml:space="preserve" subscript="x">δ</ml:id> </ml:bounds> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="44"/> </region> <region region-id="221" left="84" top="1809" width="90" 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<ml:apply> <ml:mult/> <ml:real>2.1</ml:real> <ml:id xml:space="preserve">pu</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="12" left="18" top="146.25" width="416.25" height="72" align-x="21" align-y="156" 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"> A synchronous generator with the parameters given above is operating with constant mechanical torque, Tm, equal to 0.7 pu and delivering power to an infinite bus.<sp count="2"/>The electrical power (torque) generated by the generator goes to zero for a period of time, tc, (clearing time).<sp count="2"/>The clearing angle, <f family="Symbol" charset="2">d</f> c, is the angle of the rotor at tc.<sp count="2"/>The critical clearing time, tcr, is the maximum clearing time for stable operation.<sp count="2"/>The critical clearing angle, <f family="Symbol" charset="2">d</f> cr, is the angle of the rotor at tcr. </text> </region> <region region-id="32" left="24" top="231" width="51" height="16.5" align-x="42.75" align-y="240" 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="m">T</ml:id> <ml:apply> <ml:mult/> <ml:real>0.7</ml:real> <ml:id xml:space="preserve">pu</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="16" left="18" top="266.25" width="58.5" height="12" align-x="33.75" align-y="276" 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"> Assume that: </text> </region> <region region-id="20" left="54" top="297" width="152.25" height="57" align-x="89.25" align-y="306" 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="e">T</ml:id> <ml:sequence> <ml:id xml:space="preserve">δ</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> <ml:program> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="max">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">t</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:greaterOrEqual/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve" subscript="c">t</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="max">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:ifThen> </ml:program> </ml:apply> </math> <rendering item-idref="8"/> </region> <region region-id="22" left="6" top="372.75" width="57" height="13.5" align-x="9.75" 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="false"> II. Desired </text> </region> <region region-id="24" left="36" top="410.25" width="23.25" height="12" align-x="43.5" align-y="420" 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"> Part I </text> </region> <region region-id="26" left="54" top="440.25" width="284.25" height="72" align-x="65.25" align-y="450" 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"> Using analytical methods (not simulation) calcuate the: <tab/>1. critical clearing angle, <f family="Symbol" charset="2">d</f> cr, <tab/>2. critical clearing time, tcr, <tab/>3. the maximum swing angle, <f family="Symbol" charset="2">d</f> maxswing (with tc = 0.8 tcr), and <tab/>4. the minimum swing angle, minswing (with tc = 0.8 tcr). </text> </region> <region region-id="27" left="12" top="630.75" width="65.25" height="13.5" align-x="15.75" align-y="642" 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"> III. Solution </text> </region> <region region-id="36" left="36" top="656.25" width="183" height="15" align-x="37.5" align-y="666" 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"> 1. Calculate the critical clearing angle, δcr. </text> </region> <region region-id="39" left="54" top="686.25" width="161.25" height="15" align-x="57" align-y="696" 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"> a. Calculate the initial rotor angle, δ0. </text> </region> <region region-id="40" left="72" top="711" width="99" height="16.5" align-x="111.75" align-y="720" 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:id xml:space="preserve" subscript="e">T</ml:id> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="max">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve" subscript="0">δ</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="9"/> </region> <region region-id="110" left="72" top="742.5" width="75" height="35.25" align-x="87" align-y="762" 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="0">δ</ml:id> <ml:apply> <ml:id xml:space="preserve">asin</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="m">T</ml:id> <ml:id xml:space="preserve" subscript="max">T</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="108" left="246" top="753" width="52.5" height="16.5" align-x="260.25" align-y="762" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="0">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.33983690945412193</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="11"/> </region> <region region-id="109" left="348" top="753" width="63.75" height="16.5" align-x="362.25" align-y="762" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="0">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>19.471220634490692</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="12"/> </region> <region region-id="47" left="54" top="788.25" width="78" height="15" align-x="57" align-y="798" 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"> b. Calculate δmax. </text> </region> <region region-id="51" left="144" top="788.25" width="219.75" height="12" align-x="151.5" align-y="798" 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"> The following is only valid for round rotor machines. </text> </region> <region region-id="56" left="78" top="819" width="60" height="16.5" align-x="103.5" align-y="828" 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">δ</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve" subscript="0">δ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="55" left="240" top="819" width="67.5" height="16.5" align-x="264.75" align-y="828" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="max">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.8017557441356713</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="14"/> </region> <region region-id="59" left="348" top="819" width="78.75" height="16.5" align-x="372.75" align-y="828" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="max">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>160.52877936550931</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="15"/> </region> <region region-id="63" left="54" top="854.25" width="72" height="15" align-x="55.5" align-y="864" 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"> c. Calculate δ.cr. </text> </region> <region region-id="70" left="78" top="884.25" width="54.75" height="12" align-x="88.5" align-y="894" 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"> Initial guess: </text> </region> <region region-id="87" left="162" top="874.5" width="65.25" height="31.5" align-x="176.25" align-y="894" 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="x">δ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="max">δ</ml:id> <ml:id xml:space="preserve" subscript="0">δ</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="86" left="258" top="885" width="56.25" height="16.5" align-x="271.5" align-y="894" 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="x">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.5707963267948966</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="17"/> </region> <region region-id="88" left="354" top="885" width="47.25" height="16.5" align-x="367.5" align-y="894" 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="x">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <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="18"/> </region> <region region-id="71" left="72" top="921" width="24" height="12.75" align-x="83.25" align-y="930" show-border="false" show-highlight="false" 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/math30">Given</ml:id> </math> <rendering item-idref="19"/> </region> <region region-id="77" left="102" top="939.75" width="174.75" height="48.75" align-x="150" align-y="966" 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" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">δ</ml:id> </ml:boundVars> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve" subscript="0">δ</ml:id> <ml:id xml:space="preserve" subscript="x">δ</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">δ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="max">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve" subscript="x">δ</ml:id> <ml:id xml:space="preserve" subscript="max">δ</ml:id> </ml:bounds> </ml:apply> </ml:apply> </math> <rendering item-idref="20"/> </region> <region region-id="93" left="72" top="1017" width="59.25" height="16.5" align-x="90" align-y="1026" 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="cr">δ</ml:id> <ml:apply> <ml:Find auto-method="true" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:id xml:space="preserve" subscript="x">δ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="92" left="222" top="1017" width="60" height="16.5" align-x="239.25" align-y="1026" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.69327172026567</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="91" left="330" top="1017" width="66.75" height="16.5" align-x="347.25" align-y="1026" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>97.017323140079426</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="23"/> </region> <region region-id="96" left="48" top="1064.25" width="159" height="12" align-x="49.5" align-y="1074" 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"> 2. Calcuate the critical clearing time: </text> </region> <region region-id="100" left="84" top="1084.5" width="102" height="37.5" align-x="101.25" align-y="1110" 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:id xml:space="preserve" subscript="cr">δ</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="eB">ω</ml:id> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">H</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve" subscript="cr">t</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="0">δ</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="24"/> </region> <region region-id="104" left="240" top="1100.25" width="109.5" height="12" align-x="249.75" align-y="1110" 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"> From Session 20 page 9. </text> </region> <region region-id="105" left="84" top="1134.75" width="109.5" height="33" align-x="100.5" align-y="1152" 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="cr">t</ml:id> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">H</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="eB">ω</ml:id> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="cr">δ</ml:id> <ml:id xml:space="preserve" subscript="0">δ</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="107" left="240" top="1149" width="59.25" height="16.5" align-x="255.75" align-y="1158" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="cr">t</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.22646656448963631</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="second"/> </unitMonomial> </unitedValue> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="4" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="26"/> </region> <region region-id="112" left="48" top="1262.25" width="238.5" height="15" align-x="49.5" align-y="1272" 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"> 3. Calculate the maximum swing angle with tc = 0.8 tcr. </text> </region> <region region-id="113" left="72" top="1299" width="45.75" height="16.5" align-x="85.5" align-y="1308" 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="c">t</ml:id> <ml:apply> <ml:mult/> <ml:real>0.8</ml:real> <ml:id xml:space="preserve" subscript="cr">t</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="114" left="252" top="1299" width="56.25" height="16.5" align-x="264.75" align-y="1308" 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="c">t</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.18117325159170905</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="second"/> </unitMonomial> </unitedValue> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="4" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="28"/> </region> <region region-id="119" left="72" top="1336.5" width="97.5" height="37.5" align-x="87" align-y="1362" 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="c">δ</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="eB">ω</ml:id> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">H</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve" subscript="c">t</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="0">δ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="120" left="252" top="1353" width="57" height="16.5" align-x="266.25" align-y="1362" 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="c">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.2060351883735128</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="30"/> </region> <region region-id="153" left="342" top="1353" width="63.75" height="16.5" align-x="356.25" align-y="1362" 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="c">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>69.1007262380675</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="31"/> </region> <region region-id="146" left="72" top="1406.25" width="54.75" height="12" align-x="82.5" align-y="1416" 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"> Initial guess: </text> </region> <region region-id="147" left="156" top="1396.5" width="68.25" height="31.5" align-x="170.25" 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:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="x">δ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="max">δ</ml:id> <ml:id xml:space="preserve" subscript="cr">δ</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="150" left="252" top="1407" width="56.25" height="16.5" align-x="265.5" 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:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="x">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.2475137322006704</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="33"/> </region> <region region-id="152" left="342" top="1407" width="67.5" height="16.5" align-x="355.5" 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:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="x">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>128.77305125279435</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="34"/> </region> <region region-id="176" left="72" top="1449" width="24" height="12.75" align-x="83.25" 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:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="35"/> </region> <region region-id="177" left="102" top="1467.75" width="164.25" height="48.75" align-x="150" 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/math30"> <ml:equal/> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">δ</ml:id> </ml:boundVars> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve" subscript="0">δ</ml:id> <ml:id xml:space="preserve" subscript="c">δ</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">δ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="max">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve" subscript="c">δ</ml:id> <ml:id xml:space="preserve" subscript="x">δ</ml:id> </ml:bounds> </ml:apply> </ml:apply> </math> <rendering item-idref="36"/> </region> <region region-id="178" left="72" top="1545" width="90.75" height="16.5" align-x="121.5" align-y="1554" 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="maxswing">δ</ml:id> <ml:apply> <ml:Find auto-method="true" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:id xml:space="preserve" subscript="x">δ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="179" left="216" top="1545" width="91.5" height="16.5" align-x="264.75" align-y="1554" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="maxswing">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.6513172729909318</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="38"/> </region> <region region-id="180" left="324" top="1545" width="98.25" height="16.5" align-x="372.75" align-y="1554" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="maxswing">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>94.6135103794328</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="39"/> </region> <region region-id="200" left="66" top="1598.25" width="238.5" height="15" align-x="67.5" align-y="1608" 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"> 3. Calculate the maximum swing angle with tc = 0.8 tcr. </text> </region> <region region-id="215" left="84" top="1670.25" width="54.75" height="12" align-x="94.5" align-y="1680" 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"> Initial guess: </text> </region> <region region-id="216" left="168" top="1664.25" width="36.75" height="27.75" align-x="182.25" align-y="1680" 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/math30"> <ml:id xml:space="preserve" subscript="x">δ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>4</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="40"/> </region> <region region-id="217" left="264" top="1671" width="61.5" height="16.5" align-x="277.5" align-y="1680" 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="x">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-0.78539816339744828</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="41"/> </region> <region region-id="218" left="354" top="1671" width="52.5" height="16.5" align-x="367.5" align-y="1680" 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="x">δ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-45</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="42"/> </region> <region region-id="219" left="84" top="1713" width="24" height="12.75" align-x="95.25" align-y="1722" show-border="false" show-highlight="false" 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/math30">Given</ml:id> </math> <rendering item-idref="43"/> </region> <region region-id="220" left="114" top="1731.75" width="155.25" height="48.75" align-x="258" align-y="1758" 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" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">δ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="max">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="m">T</ml:id> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve" subscript="maxswing">δ</ml:id> <ml:id xml:space="preserve" subscript="x">δ</ml:id> </ml:bounds> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="44"/> </region> <region region-id="221" left="84" top="1809" width="90" height="16.5" align-x="132.75" align-y="1818" 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="minswing">δ</ml:id> <ml:apply> <ml:Find auto-method="true" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:id xml:space="preserve" subscript="x">δ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="45"/> </region> <region region-id="222" left="228" top="1809" width="96" height="16.5" align-x="276" align-y="1818" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id 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