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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">E</ml:id> <ml:id xml:space="preserve" subscript="qp">E</ml:id> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="365" left="222" top="436.5" width="63" height="35.25" align-x="242.25" align-y="456" 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="1f">T</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="sys">V</ml:id> <ml:id xml:space="preserve" subscript="x">E</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="df">X</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="279" left="366" top="447" width="52.5" height="16.5" align-x="385.5" align-y="456" 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="1f">T</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>3.2988721804511272</ml:real> </result> </ml:eval> </math> <rendering item-idref="27"/> </region> <region region-id="366" left="222" top="478.5" width="113.25" height="41.25" align-x="242.25" align-y="504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2f">T</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve" subscript="sys">V</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="df">X</ml:id> <ml:id xml:space="preserve" subscript="qf">X</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="df">X</ml:id> <ml:id xml:space="preserve" subscript="qf">X</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="284" left="366" top="495" width="57.75" height="16.5" align-x="385.5" align-y="504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2f">T</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>-0.97117794486215514</ml:real> </result> </ml:eval> </math> <rendering item-idref="29"/> </region> <region region-id="12" left="18" top="536.25" width="416.25" height="72" align-x="21" align-y="546" 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 0.77 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, γc, 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, γcr, is the angle of the rotor at tcr. </text> </region> <region region-id="32" left="24" top="621" width="55.5" height="16.5" align-x="42.75" align-y="630" 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>0.77</ml:real> <ml:id xml:space="preserve">pu</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="16" left="18" top="656.25" width="58.5" height="12" align-x="33.75" 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"> Assume that: </text> </region> <region region-id="20" left="54" top="687" width="209.25" height="57" align-x="89.25" align-y="696" 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:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1ss">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2ss">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </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:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:ifThen> </ml:program> </ml:apply> </math> <rendering item-idref="31"/> </region> <region region-id="22" left="6" top="762.75" width="57" height="13.5" align-x="9.75" align-y="774" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> II. Desired </text> </region> <region region-id="24" left="36" top="800.25" width="23.25" height="12" align-x="43.5" align-y="810" 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="830.25" width="268.5" height="72" align-x="65.25" align-y="840" 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, γcr, <tab/>2. critical clearing time, tcr, <tab/>3. the maximum swing angle, γmaxswing (with tc = tcr), and <tab/>4. the minimum swing angle,γminswing (with tc = tcr). </text> </region> <region region-id="27" left="12" top="954.75" width="65.25" height="13.5" align-x="15.75" align-y="966" 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="980.25" width="183" height="15" align-x="37.5" align-y="990" 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, γcr. </text> </region> <region region-id="39" left="54" top="1010.25" width="161.25" height="15" align-x="57" align-y="1020" 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, γ0. </text> </region> <region region-id="40" left="72" top="1035" width="171" height="16.5" align-x="111.75" align-y="1044" 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="m">T</ml:id> <ml:id xml:space="preserve" subscript="e">T</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1ss">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve" subscript="0">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2ss">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve" subscript="0">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="32"/> </region> <region region-id="241" left="72" top="1076.25" width="51.75" height="12" align-x="82.5" align-y="1086" 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="240" left="150" top="1066.5" width="74.25" height="35.25" align-x="165.75" align-y="1086" 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">γ</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="1ss">T</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="239" left="246" top="1077" width="57.75" height="16.5" align-x="261" align-y="1086" 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="x">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.61351463657140948</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="34"/> </region> <region region-id="238" left="348" top="1077" width="64.5" height="16.5" align-x="363" align-y="1086" 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="x">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>35.15179934504431</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="35"/> </region> <region region-id="236" left="72" top="1107" width="24" height="12.75" align-x="83.25" align-y="1116" 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="36"/> </region> <region region-id="257" left="126" top="1125" width="147.75" height="16.5" align-x="144" align-y="1134" 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="m">T</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1ss">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2ss">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="37"/> </region> <region region-id="260" left="78" top="1155" width="59.25" height="16.5" align-x="94.5" align-y="1164" 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">γ</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">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="259" left="252" top="1155" width="58.5" height="16.5" align-x="267.75" align-y="1164" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.47843839793921294</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="39"/> </region> <region region-id="258" left="354" top="1155" width="65.25" height="16.5" align-x="369.75" align-y="1164" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>27.412500958917484</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="40"/> </region> <region region-id="47" left="54" top="1250.25" width="78" height="15" align-x="57" align-y="1260" 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="272" left="78" top="1280.25" width="51.75" height="12" align-x="88.5" align-y="1290" 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="269" left="150" top="1281" width="51.75" height="16.5" align-x="165.75" align-y="1290" 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">γ</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve" subscript="0">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="243" left="240" top="1281" width="57.75" height="16.5" align-x="255" align-y="1290" 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="x">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.6631542556505803</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="42"/> </region> <region region-id="244" left="348" top="1281" width="69" height="16.5" align-x="363" align-y="1290" 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="x">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>152.58749904108251</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="43"/> </region> <region region-id="251" left="78" top="1305" width="24" height="12.75" align-x="89.25" align-y="1314" 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="44"/> </region> <region region-id="267" left="132" top="1323" width="137.25" height="16.5" align-x="150" align-y="1332" 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="m">T</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="45"/> </region> <region region-id="264" left="78" top="1353" width="69.75" height="16.5" align-x="105" 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="max">γ</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">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="265" left="252" top="1353" width="69" height="16.5" align-x="278.25" align-y="1362" 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="max">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.9935422442815058</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="47"/> </region> <region region-id="266" left="354" top="1353" width="80.25" height="16.5" align-x="380.25" align-y="1362" 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="max">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>171.51733639145078</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="48"/> </region> <region region-id="63" left="54" top="1412.25" width="72" height="15" align-x="55.5" align-y="1422" 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 γ.cr. </text> </region> <region region-id="268" left="78" top="1442.25" width="54.75" height="12" align-x="88.5" align-y="1452" 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="1432.5" width="69.75" height="31.5" align-x="177.75" align-y="1452" 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">γ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="max">γ</ml:id> <ml:id xml:space="preserve" subscript="0">γ</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="49"/> </region> <region region-id="86" left="258" top="1443" width="57.75" height="16.5" align-x="273" align-y="1452" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.7359903211103593</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="50"/> </region> <region region-id="88" left="354" top="1443" width="64.5" height="16.5" align-x="369" align-y="1452" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>99.464918675184123</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="51"/> </region> <region region-id="71" left="72" top="1473" width="24" height="12.75" align-x="83.25" align-y="1482" 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="52"/> </region> <region region-id="77" left="102" top="1491.75" width="236.25" height="48.75" align-x="152.25" align-y="1518" 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">γ</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">γ</ml:id> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">γ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </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">γ</ml:id> <ml:id xml:space="preserve" subscript="max">γ</ml:id> </ml:bounds> </ml:apply> </ml:apply> </math> <rendering item-idref="53"/> </region> <region region-id="93" left="72" top="1569" width="62.25" height="16.5" align-x="91.5" align-y="1578" 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">γ</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">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="54"/> </region> <region region-id="92" left="222" top="1569" width="52.5" height="16.5" align-x="240.75" align-y="1578" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.19957086053264</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="55"/> </region> <region region-id="91" left="330" top="1569" width="72.75" height="16.5" align-x="348.75" align-y="1578" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>126.02612704847888</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="56"/> </region> <region region-id="96" left="48" top="1628.25" width="159" height="12" align-x="49.5" align-y="1638" 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="1648.5" width="105" height="37.5" align-x="102.75" align-y="1674" 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">γ</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">γ</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="57"/> </region> <region region-id="104" left="240" top="1664.25" width="109.5" height="12" align-x="249.75" align-y="1674" 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="1698.75" width="112.5" height="33" align-x="100.5" align-y="1716" 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">γ</ml:id> <ml:id xml:space="preserve" subscript="0">γ</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="58"/> </region> <region region-id="107" left="240" top="1713" width="59.25" height="16.5" align-x="255.75" align-y="1722" 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>0.28396556602848355</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="true" 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="59"/> </region> <region region-id="112" left="48" top="1868.25" width="225" height="15" align-x="49.5" align-y="1878" 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 =<sp count="2"/>tcr. </text> </region> <region region-id="113" left="72" top="1893" width="31.5" height="16.5" align-x="85.5" align-y="1902" 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:id xml:space="preserve" subscript="cr">t</ml:id> </ml:define> </math> <rendering item-idref="60"/> </region> <region region-id="114" left="252" top="1893" width="51.75" height="16.5" align-x="264.75" align-y="1902" 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>0.28396556602848355</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="61"/> </region> <region region-id="119" left="72" top="1924.5" width="100.5" height="37.5" align-x="88.5" align-y="1950" 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">γ</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">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="62"/> </region> <region region-id="120" left="252" top="1941" width="49.5" height="16.5" align-x="267.75" align-y="1950" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.19957086053264</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="63"/> </region> <region region-id="153" left="342" top="1941" width="69.75" height="16.5" align-x="357.75" align-y="1950" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>126.02612704847888</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="64"/> </region> <region region-id="146" left="72" top="1994.25" width="54.75" height="12" align-x="82.5" align-y="2004" 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="1984.5" width="72.75" height="31.5" align-x="171.75" align-y="2004" 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">γ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="max">γ</ml:id> <ml:id xml:space="preserve" subscript="cr">γ</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="65"/> </region> <region region-id="150" left="252" top="1995" width="57.75" height="16.5" align-x="267" align-y="2004" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.5965565524070726</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="66"/> </region> <region region-id="152" left="342" top="1995" width="69" height="16.5" align-x="357" align-y="2004" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>148.77173171996481</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="67"/> </region> <region region-id="176" left="72" top="2031" width="24" height="12.75" align-x="83.25" align-y="2040" 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="68"/> </region> <region region-id="177" left="102" top="2049.75" width="222.75" height="48.75" align-x="152.25" align-y="2076" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml: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">γ</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">γ</ml:id> <ml:id xml:space="preserve" subscript="c">γ</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">γ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real font="0">2</ml:real> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </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">γ</ml:id> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:bounds> </ml:apply> </ml:apply> </math> <rendering item-idref="69"/> </region> <region region-id="178" left="72" top="2115" width="93.75" height="16.5" align-x="123" 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:id xml:space="preserve" subscript="maxswing">γ</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">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="70"/> </region> <region region-id="179" left="216" top="2115" width="93" height="16.5" align-x="266.25" align-y="2124" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.9935422211536014</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="71"/> </region> <region region-id="180" left="324" top="2115" width="104.25" height="16.5" align-x="374.25" align-y="2124" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>171.51733506631948</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="72"/> </region> <region region-id="200" left="66" top="2168.25" width="238.5" height="15" align-x="67.5" align-y="2178" 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 =<sp count="2"/>tcr. </text> </region> <region region-id="215" left="84" top="2198.25" width="54.75" height="12" align-x="94.5" align-y="2208" 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="2192.25" width="38.25" height="27.75" align-x="183.75" align-y="2208" 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">γ</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="73"/> </region> <region region-id="217" left="264" top="2199" width="63" height="16.5" align-x="279" align-y="2208" 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">γ</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="74"/> </region> <region region-id="218" left="354" top="2199" width="54" height="16.5" align-x="369" align-y="2208" 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">γ</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="75"/> </region> <region region-id="219" left="84" top="2229" width="24" height="12.75" align-x="95.25" align-y="2238" 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="76"/> </region> <region region-id="220" left="114" top="2247.75" width="214.5" height="48.75" align-x="317.25" align-y="2274" 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">γ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </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">γ</ml:id> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:bounds> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="77"/> </region> <region region-id="221" left="84" top="2319" width="93" height="16.5" align-x="134.25" align-y="2328" 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">γ</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">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="78"/> </region> <region region-id="367" left="222" top="2319" width="97.5" height="16.5" align-x="271.5" 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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>3.4</ml:real> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> 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<ml:real>-0.97117794486215514</ml:real> </result> </ml:eval> </math> <rendering item-idref="29"/> </region> <region region-id="12" left="18" top="536.25" width="416.25" height="72" align-x="21" align-y="546" 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 0.77 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, γc, 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, γcr, is the angle of the rotor at tcr. </text> </region> <region region-id="32" left="24" top="621" width="55.5" height="16.5" align-x="42.75" align-y="630" 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>0.77</ml:real> <ml:id xml:space="preserve">pu</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="16" left="18" top="656.25" width="58.5" height="12" align-x="33.75" 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"> Assume that: </text> </region> <region region-id="20" left="54" top="687" width="209.25" height="57" align-x="89.25" align-y="696" 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:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1ss">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2ss">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </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:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:ifThen> </ml:program> </ml:apply> </math> <rendering item-idref="31"/> </region> <region region-id="22" left="6" top="762.75" width="57" height="13.5" align-x="9.75" align-y="774" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> II. Desired </text> </region> <region region-id="24" left="36" top="800.25" width="23.25" height="12" align-x="43.5" align-y="810" 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="830.25" width="268.5" height="72" align-x="65.25" align-y="840" 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, γcr, <tab/>2. critical clearing time, tcr, <tab/>3. the maximum swing angle, γmaxswing (with tc = tcr), and <tab/>4. the minimum swing angle,γminswing (with tc = tcr). </text> </region> <region region-id="27" left="12" top="954.75" width="65.25" height="13.5" align-x="15.75" align-y="966" 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="980.25" width="183" height="15" align-x="37.5" align-y="990" 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, γcr. </text> </region> <region region-id="39" left="54" top="1010.25" width="161.25" height="15" align-x="57" align-y="1020" 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, γ0. </text> </region> <region region-id="40" left="72" top="1035" width="171" height="16.5" align-x="111.75" align-y="1044" 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="m">T</ml:id> <ml:id xml:space="preserve" subscript="e">T</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1ss">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve" subscript="0">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2ss">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve" subscript="0">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="32"/> </region> <region region-id="241" left="72" top="1076.25" width="51.75" height="12" align-x="82.5" align-y="1086" 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="240" left="150" top="1066.5" width="74.25" height="35.25" align-x="165.75" align-y="1086" 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">γ</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="1ss">T</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="239" left="246" top="1077" width="57.75" height="16.5" align-x="261" align-y="1086" 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="x">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.61351463657140948</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="34"/> </region> <region region-id="238" left="348" top="1077" width="64.5" height="16.5" align-x="363" align-y="1086" 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="x">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>35.15179934504431</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="35"/> </region> <region region-id="236" left="72" top="1107" width="24" height="12.75" align-x="83.25" align-y="1116" 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="36"/> </region> <region region-id="257" left="126" top="1125" width="147.75" height="16.5" align-x="144" align-y="1134" 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="m">T</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1ss">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2ss">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="37"/> </region> <region region-id="260" left="78" top="1155" width="59.25" height="16.5" align-x="94.5" align-y="1164" 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">γ</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">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="259" left="252" top="1155" width="58.5" height="16.5" align-x="267.75" align-y="1164" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.47843839793921294</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="39"/> </region> <region region-id="258" left="354" top="1155" width="65.25" height="16.5" align-x="369.75" align-y="1164" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>27.412500958917484</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="40"/> </region> <region region-id="47" left="54" top="1250.25" width="78" height="15" align-x="57" align-y="1260" 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="272" left="78" top="1280.25" width="51.75" height="12" align-x="88.5" align-y="1290" 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="269" left="150" top="1281" width="51.75" height="16.5" align-x="165.75" align-y="1290" 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">γ</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve" subscript="0">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="243" left="240" top="1281" width="57.75" height="16.5" align-x="255" align-y="1290" 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="x">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.6631542556505803</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="42"/> </region> <region region-id="244" left="348" top="1281" width="69" height="16.5" align-x="363" align-y="1290" 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="x">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>152.58749904108251</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="43"/> </region> <region region-id="251" left="78" top="1305" width="24" height="12.75" align-x="89.25" align-y="1314" 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="44"/> </region> <region region-id="267" left="132" top="1323" width="137.25" height="16.5" align-x="150" align-y="1332" 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="m">T</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="45"/> </region> <region region-id="264" left="78" top="1353" width="69.75" height="16.5" align-x="105" 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="max">γ</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">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="265" left="252" top="1353" width="69" height="16.5" align-x="278.25" align-y="1362" 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="max">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.9935422442815058</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="47"/> </region> <region region-id="266" left="354" top="1353" width="80.25" height="16.5" align-x="380.25" align-y="1362" 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="max">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>171.51733639145078</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="48"/> </region> <region region-id="63" left="54" top="1412.25" width="72" height="15" align-x="55.5" align-y="1422" 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 γ.cr. </text> </region> <region region-id="268" left="78" top="1442.25" width="54.75" height="12" align-x="88.5" align-y="1452" 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="1432.5" width="69.75" height="31.5" align-x="177.75" align-y="1452" 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">γ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="max">γ</ml:id> <ml:id xml:space="preserve" subscript="0">γ</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="49"/> </region> <region region-id="86" left="258" top="1443" width="57.75" height="16.5" align-x="273" align-y="1452" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.7359903211103593</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="50"/> </region> <region region-id="88" left="354" top="1443" width="64.5" height="16.5" align-x="369" align-y="1452" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>99.464918675184123</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="51"/> </region> <region region-id="71" left="72" top="1473" width="24" height="12.75" align-x="83.25" align-y="1482" 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="52"/> </region> <region region-id="77" left="102" top="1491.75" width="236.25" height="48.75" align-x="152.25" align-y="1518" 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">γ</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">γ</ml:id> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">γ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </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">γ</ml:id> <ml:id xml:space="preserve" subscript="max">γ</ml:id> </ml:bounds> </ml:apply> </ml:apply> </math> <rendering item-idref="53"/> </region> <region region-id="93" left="72" top="1569" width="62.25" height="16.5" align-x="91.5" align-y="1578" 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">γ</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">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="54"/> </region> <region region-id="92" left="222" top="1569" width="52.5" height="16.5" align-x="240.75" align-y="1578" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.19957086053264</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="55"/> </region> <region region-id="91" left="330" top="1569" width="72.75" height="16.5" align-x="348.75" align-y="1578" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>126.02612704847888</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="56"/> </region> <region region-id="96" left="48" top="1628.25" width="159" height="12" align-x="49.5" align-y="1638" 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="1648.5" width="105" height="37.5" align-x="102.75" align-y="1674" 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">γ</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">γ</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="57"/> </region> <region region-id="104" left="240" top="1664.25" width="109.5" height="12" align-x="249.75" align-y="1674" 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="1698.75" width="112.5" height="33" align-x="100.5" align-y="1716" 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">γ</ml:id> <ml:id xml:space="preserve" subscript="0">γ</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="58"/> </region> <region region-id="107" left="240" top="1713" width="59.25" height="16.5" align-x="255.75" align-y="1722" 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>0.28396556602848355</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="true" 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="59"/> </region> <region region-id="112" left="48" top="1868.25" width="225" height="15" align-x="49.5" align-y="1878" 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 =<sp count="2"/>tcr. </text> </region> <region region-id="113" left="72" top="1893" width="31.5" height="16.5" align-x="85.5" align-y="1902" 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:id xml:space="preserve" subscript="cr">t</ml:id> </ml:define> </math> <rendering item-idref="60"/> </region> <region region-id="114" left="252" top="1893" width="51.75" height="16.5" align-x="264.75" align-y="1902" 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>0.28396556602848355</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="61"/> </region> <region region-id="119" left="72" top="1924.5" width="100.5" height="37.5" align-x="88.5" align-y="1950" 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">γ</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">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="62"/> </region> <region region-id="120" left="252" top="1941" width="49.5" height="16.5" align-x="267.75" align-y="1950" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.19957086053264</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="63"/> </region> <region region-id="153" left="342" top="1941" width="69.75" height="16.5" align-x="357.75" align-y="1950" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>126.02612704847888</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="64"/> </region> <region region-id="146" left="72" top="1994.25" width="54.75" height="12" align-x="82.5" align-y="2004" 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="1984.5" width="72.75" height="31.5" align-x="171.75" align-y="2004" 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">γ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="max">γ</ml:id> <ml:id xml:space="preserve" subscript="cr">γ</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="65"/> </region> <region region-id="150" left="252" top="1995" width="57.75" height="16.5" align-x="267" align-y="2004" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.5965565524070726</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="66"/> </region> <region region-id="152" left="342" top="1995" width="69" height="16.5" align-x="357" align-y="2004" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>148.77173171996481</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="67"/> </region> <region region-id="176" left="72" top="2031" width="24" height="12.75" align-x="83.25" align-y="2040" 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="68"/> </region> <region region-id="177" left="102" top="2049.75" width="222.75" height="48.75" align-x="152.25" align-y="2076" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml: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">γ</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">γ</ml:id> <ml:id xml:space="preserve" subscript="c">γ</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">γ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real font="0">2</ml:real> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </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">γ</ml:id> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:bounds> </ml:apply> </ml:apply> </math> <rendering item-idref="69"/> </region> <region region-id="178" left="72" top="2115" width="93.75" height="16.5" align-x="123" 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:id xml:space="preserve" subscript="maxswing">γ</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">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="70"/> </region> <region region-id="179" left="216" top="2115" width="93" height="16.5" align-x="266.25" align-y="2124" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.9935422211536014</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="71"/> </region> <region region-id="180" left="324" top="2115" width="104.25" height="16.5" align-x="374.25" align-y="2124" 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">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>171.51733506631948</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="72"/> </region> <region region-id="200" left="66" top="2168.25" width="238.5" height="15" align-x="67.5" align-y="2178" 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 =<sp count="2"/>tcr. </text> </region> <region region-id="215" left="84" top="2198.25" width="54.75" height="12" align-x="94.5" align-y="2208" 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="2192.25" width="38.25" height="27.75" align-x="183.75" align-y="2208" 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">γ</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="73"/> </region> <region region-id="217" left="264" top="2199" width="63" height="16.5" align-x="279" align-y="2208" 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">γ</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="74"/> </region> <region region-id="218" left="354" top="2199" width="54" height="16.5" align-x="369" align-y="2208" 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">γ</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="75"/> </region> <region region-id="219" left="84" top="2229" width="24" height="12.75" align-x="95.25" align-y="2238" 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="76"/> </region> <region region-id="220" left="114" top="2247.75" width="214.5" height="48.75" align-x="317.25" align-y="2274" 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">γ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2f">T</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </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">γ</ml:id> <ml:id xml:space="preserve" subscript="x">γ</ml:id> </ml:bounds> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="77"/> </region> <region region-id="221" left="84" top="2319" width="93" height="16.5" align-x="134.25" align-y="2328" 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">γ</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">γ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="78"/> </region> <region region-id="367" left="222" top="2319" width="97.5" height="16.5" align-x="271.5" align-y="2328" 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="minswing">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">rad</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-1.7365316535285953</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="radian"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="79"/> </region> <region region-id="368" left="330" top="2319" width="104.25" height="16.5" align-x="379.5" align-y="2328" 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="minswing">γ</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>-99.495934738062658</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="80"/> </region> </regions> <binaryContent> <item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4yQwW7CMAyG7aZbS9etu3AhSOUZeIIdENph2iT2ACiUAkVUTF2ROPLm3W8n u+w0R3b+Ot/vRsmJiJGfyEy1QS228/W+c1+H9e7cta5PSGKMHGl74XqnLZog07b62Bzrqvet ld8omqGYMF+8hWr5yty1+V7q6FjZKfKNA+tjwCKe/r/7OzzpO1fV84XRIxvAkp+JLQ+imG0E U8kRWwNryYZtPEgnBkMDxB2Y2wuO7oURV6IMRCqMwCOZI/YMjNofwKg9B6P2R/0XxJMyEIUw AvvrFu+Xtu6ayp38g6R67VfklhosE14uC6/5J3IpPwAAAP//AwARvY6sxgEAAA==</item> <item 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