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push-down="false" lock-width="true"> Metal Parameters: Aluminum characteristic thermal length,<sp count="2"/> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> ~ 25 - 200um.<sp count="2"/>Lines > <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f>are thermally "long". </text> </region> <region region-id="3772" left="30" top="922.5" width="96" height="17.25" align-x="57" align-y="936" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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<ml:real>4.5</ml:real> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">Rds_O2A</ml:id> </ml:apply> <ml:id xml:space="preserve">watt</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="125"/> </region> <region region-id="4406" left="276" top="1202.25" width="195.75" height="23.25" align-x="322.5" align-y="1218" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="load">P</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Ppk</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="tri">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:indexer/> <ml:parens> <ml:apply> <ml:vectorize/> <ml:parens> <ml:apply> <ml:not/> <ml:id xml:space="preserve">Ppk</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="htr">P</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="126"/> </region> <region region-id="2438" left="84" top="1262.25" width="387.75" height="36" align-x="99" align-y="1272" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Solution for <f family="Arial" charset="0"> <inlineAttr line-through="false"> Green's Function </inlineAttr> </f> to <f family="Arial" charset="0"> <inlineAttr line-through="false">Transient Temperature Equation for Rectangle</inlineAttr> </f> <inlineAttr font-weight="normal" underline="false" line-through="false"> <f family="Arial" charset="0">S</f> </inlineAttr> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" underline="false" line-through="false">urface Source Model Predicts higher temp and faster transients.</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" underline="false" line-through="false">For x,<sp count="2"/>sources have edge Lox and width W. 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</region> <region region-id="5083" left="24" top="1412.25" width="221.25" height="15.75" align-x="113.25" align-y="1422" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">PSources</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="pk">P</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="tri">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:real>12</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="S6">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="131"/> </region> <region region-id="3880" left="186" top="1448.25" width="250.5" height="12" align-x="204.75" align-y="1458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0"> <inlineAttr line-through="false">Greens Transient Temperature Rise for Sources, </inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-style="normal" line-through="false">D</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false">T</inlineAttr> </f> <inlineAttr line-through="false"> <f family="Arial" charset="0">o</f> </inlineAttr> </text> </region> <region region-id="5080" left="54" top="1490.25" width="197.25" height="21.75" align-x="96.75" align-y="1506" 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">ISqkρCp</ml:id> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="tong">k</ml:id> <ml:id xml:space="preserve" subscript="ambK">T</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ρCp</ml:id> <ml:id xml:space="preserve" subscript="ambK">T</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:real>-1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="132"/> </region> <region region-id="5079" left="48" top="1520.25" width="214.5" height="21.75" align-x="90.75" align-y="1536" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="true"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">ISqkρCp</ml:id> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="tong">k</ml:id> <ml:id xml:space="preserve" subscript="ambK">T</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ρCp</ml:id> <ml:apply> <ml:plus/> <ml:real>273</ml:real> <ml:real>180</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:real>-1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="133"/> </region> <region region-id="3878" left="6" top="1561.5" width="540" height="58.5" align-x="78" align-y="1596" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="os">ΔT</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ISqkρCp</ml:id> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:id xml:space="preserve">PSources</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="134"/> </region> <region region-id="4997" left="6" top="1624.5" width="621.75" height="87.75" align-x="147" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="o">ΔT</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve" subscript="x">L</ml:id> <ml:id xml:space="preserve">W</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:id xml:space="preserve">H</ml:id> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ISqkρCp</ml:id> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="load">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve" subscript="x">L</ml:id> <ml:id xml:space="preserve">W</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:id xml:space="preserve">H</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="135"/> </region> <region region-id="3563" left="66" top="1748.25" width="414" height="48" align-x="93.75" align-y="1758" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0"> <inlineAttr line-through="false">Remove Heat Equation Nonlinearity from Temperature Dependence of Conductivity</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false">Kirchhoff Transformation:<sp count="2"/> </inlineAttr> </f> K, D must have value at Tamb <f family="Arial" charset="0"> <inlineAttr font-weight="normal" underline="false" line-through="false">Tk Kirch is the Inverse of 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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">Θ</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="s">T</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="s">λ</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">T</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">λ</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">Ts</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="136"/> </region> <region region-id="2568" left="360" top="1806.75" width="136.5" height="41.25" align-x="436.5" align-y="1830" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="Kirch34">Tk</ml:id> <ml:boundVars> <ml:id xml:space="preserve">Θ</ml:id> <ml:id xml:space="preserve" subscript="o">T</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>27</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve" subscript="o">T</ml:id> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="o">T</ml:id> </ml:apply> <ml:id xml:space="preserve">Θ</ml:id> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="137"/> </region> <region region-id="3565" left="90" top="1802.25" width="97.5" height="45.75" align-x="123" align-y="1836" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="Si">λ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">T</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="300">K</ml:id> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:div/> <ml:real>300</ml:real> <ml:id xml:space="preserve">T</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="138"/> </region> <region region-id="4357" left="132" top="1868.25" width="333.75" height="12" align-x="150" 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="true"> <f family="Arial" charset="0"> <inlineAttr line-through="false">Calculate Temperature versus Time.<sp count="2"/>Find Peak Temp at Given Output</inlineAttr> </f> </text> </region> <region region-id="4333" left="132" top="1898.25" width="79.5" height="12" align-x="165" align-y="1908" 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"> Evaluation Times: </text> </region> <region region-id="4334" left="222" top="1898.25" width="41.25" height="13.5" align-x="234" align-y="1908" 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="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">rt</ml:id> <ml:range> <ml:real>0</ml:real> <ml:real>60</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="139"/> </region> <region region-id="4335" left="282" top="1888.5" width="101.25" height="35.25" align-x="314.25" align-y="1908" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Time</ml:id> <ml:id xml:space="preserve">rt</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>15</ml:real> <ml:id xml:space="preserve" subscript="endμs">time</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>700</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">rt</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="140"/> </region> <region region-id="4336" left="402" top="1894.5" width="58.5" height="18.75" align-x="432.75" align-y="1908" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Time</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-9</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="141"/> </region> <region region-id="4326" left="132" top="1926.75" width="345" height="43.5" align-x="178.5" 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:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="MDj">aaaT</ml:id> <ml:id xml:space="preserve">rt</ml:id> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve" subscript="o">ΔT</ml:id> <ml:sequence> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="x">L</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="x">L</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:real>0</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Time</ml:id> <ml:id xml:space="preserve">rt</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>6</ml:real> <ml:real>6</ml:real> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="142"/> </region> <region region-id="4327" left="132" top="1982.25" width="235.5" height="19.5" align-x="184.5" align-y="1992" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="This variable is undefined." optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="MDKj">aaaT</ml:id> <ml:id xml:space="preserve">rt</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="Kirch34">Tk</ml:id> <ml:sequence> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="MDj">T</ml:id> <ml:id xml:space="preserve">rt</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="ambK">T</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="ambK">T</ml:id> </ml:sequence> </ml:apply> <ml:real font="0">273</ml:real> </ml:apply> </ml:parens> </ml:define> </math> <rendering item-idref="143"/> </region> <region region-id="4328" left="384" top="1982.25" width="72" height="15.75" align-x="441.75" align-y="1992" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="This variable is undefined." optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">max</ml:id> <ml:id xml:space="preserve" subscript="MDKj">T</ml:id> </ml:apply> </ml:eval> </math> <rendering item-idref="144"/> </region> <region region-id="5058" left="126" top="2018.25" width="137.25" height="15.75" align-x="249" align-y="2028" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="This variable is undefined." optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">match</ml:id> <ml:sequence> <ml:apply> <ml:id xml:space="preserve">max</ml:id> <ml:id xml:space="preserve" subscript="MDKj">T</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="MDKj">T</ml:id> </ml:sequence> </ml:apply> </ml:eval> </math> <rendering item-idref="145"/> </region> <region region-id="4287" left="132" top="2054.25" width="284.25" height="12" align-x="141" align-y="2064" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0"> <inlineAttr line-through="false">DMOS Out 6 FE 24W Short and Snub vs Greens ZMetal Snub</inlineAttr> </f> </text> </region> <region region-id="4289" left="396" top="2084.25" width="153.75" height="15.75" align-x="424.5" align-y="2094" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="This variable is undefined." optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="pk3">aR</ml:id> <ml:apply> <ml:id xml:space="preserve">match</ml:id> <ml:sequence> <ml:apply> <ml:id xml:space="preserve">max</ml:id> <ml:id xml:space="preserve" subscript="MDKj">T</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="MDKj">T</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="146"/> </region> <region region-id="4290" left="396" top="2102.25" width="105.75" height="21.75" align-x="423.75" align-y="2112" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="This variable is undefined." optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">atpk3</ml:id> <ml:apply> <ml:mult/> <ml:real>1000</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Time</ml:id> <ml:parens> <ml:apply> <ml:indexer/> <ml:id 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xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:real>11</ml:real> </ml:apply> <ml:apply> <ml:mult/> <ml:real>5.1</ml:real> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="93"/> </region> <region region-id="4411" left="144" top="774.75" width="262.5" height="25.5" align-x="144" align-y="786" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Times New Roman" charset="0" size="12"> <inlineAttr line-through="false"> <c val="#000">Material Constants: Silicon, Aluminum, Pedestal</c> </inlineAttr> </f> <f family="Times New Roman" charset="0"> <c val="#000">Thermal Conductivities Heat Capacity, cp, Thermal<sp count="2"/>Diffusivity, D</c> </f> </text> </region> <region region-id="1457" left="30" top="816.75" width="108.75" height="29.25" align-x="97.5" align-y="834" 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="avg">cond_silicon</ml:id> <ml:apply> <ml:mult/> <ml:real>0.8</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">watt</ml:id> <ml:id xml:space="preserve">cm</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="94"/> </region> <region region-id="1523" left="180" top="824.25" width="80.25" height="15.75" align-x="191.25" align-y="834" 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">k</ml:id> <ml:id xml:space="preserve" subscript="avg">cond_silicon</ml:id> </ml:define> </math> <rendering item-idref="95"/> </region> <region region-id="3435" left="276" top="824.25" width="52.5" height="15.75" align-x="288.75" align-y="834" 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="j">z</ml:id> <ml:apply> <ml:mult/> <ml:real>1.55</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="96"/> </region> <region region-id="1761" left="342" top="820.5" width="130.5" height="17.25" align-x="403.5" align-y="834" 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">cond_si_500C</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>0.41</ml:real> <ml:id xml:space="preserve">watt</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">cm</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="97"/> </region> <region region-id="14" left="30" top="850.5" width="108" height="17.25" align-x="56.25" align-y="864" 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">cp_si</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">watt</ml:id> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">cm</ml:id> <ml:real>-3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="98"/> </region> <region region-id="1524" left="180" top="850.5" width="89.25" height="17.25" align-x="198.75" align-y="864" 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">Ksi</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>0.87</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">cm</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">sec</ml:id> <ml:real>-1</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="99"/> </region> <region region-id="1816" left="342" top="854.25" width="73.5" height="13.5" align-x="372.75" align-y="864" 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">c_ped</ml:id> <ml:apply> <ml:mult/> <ml:real>1.5</ml:real> <ml:id xml:space="preserve">cp_si</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="100"/> </region> <region region-id="1817" left="438" top="854.25" width="85.5" height="15.75" align-x="470.25" align-y="864" 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="ambK">T</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="amb">T</ml:id> <ml:real>273</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="101"/> </region> <region region-id="3839" left="36" top="884.25" width="489.75" height="12" align-x="62.25" align-y="894" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Metal Parameters: Aluminum characteristic thermal length,<sp count="2"/> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> ~ 25 - 200um.<sp count="2"/>Lines > <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f>are thermally "long". </text> </region> <region region-id="3772" left="30" top="922.5" width="96" height="17.25" align-x="57" align-y="936" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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<ml:real>4.5</ml:real> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">Rds_O2A</ml:id> </ml:apply> <ml:id xml:space="preserve">watt</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="125"/> </region> <region region-id="4406" left="276" top="1202.25" width="195.75" height="23.25" align-x="322.5" align-y="1218" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="load">P</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Ppk</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="tri">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:indexer/> <ml:parens> <ml:apply> <ml:vectorize/> <ml:parens> <ml:apply> <ml:not/> <ml:id xml:space="preserve">Ppk</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="htr">P</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="126"/> </region> <region region-id="2438" left="84" top="1262.25" width="387.75" height="36" align-x="99" align-y="1272" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Solution for <f family="Arial" charset="0"> <inlineAttr line-through="false"> Green's Function </inlineAttr> </f> to <f 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</region> <region region-id="5083" left="24" top="1412.25" width="221.25" height="15.75" align-x="113.25" align-y="1422" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">PSources</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="pk">P</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="tri">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:real>12</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="S6">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id 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tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="os">ΔT</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ISqkρCp</ml:id> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:id xml:space="preserve">PSources</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="134"/> </region> <region region-id="4997" left="6" top="1624.5" width="621.75" height="87.75" align-x="147" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="o">ΔT</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve" subscript="x">L</ml:id> <ml:id xml:space="preserve">W</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:id xml:space="preserve">H</ml:id> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ISqkρCp</ml:id> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="load">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve" subscript="x">L</ml:id> <ml:id xml:space="preserve">W</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:id xml:space="preserve">H</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="135"/> </region> <region region-id="3563" left="66" top="1748.25" width="414" height="48" align-x="93.75" align-y="1758" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0"> <inlineAttr line-through="false">Remove Heat Equation Nonlinearity from Temperature Dependence of Conductivity</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false">Kirchhoff Transformation:<sp count="2"/> </inlineAttr> </f> K, D must have value at Tamb <f family="Arial" charset="0"> <inlineAttr font-weight="normal" underline="false" line-through="false">Tk Kirch is the Inverse of 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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">Θ</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="s">T</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="s">λ</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">T</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">λ</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">Ts</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="136"/> </region> <region region-id="2568" left="360" top="1806.75" width="136.5" height="41.25" align-x="436.5" align-y="1830" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="Kirch34">Tk</ml:id> <ml:boundVars> <ml:id xml:space="preserve">Θ</ml:id> <ml:id xml:space="preserve" subscript="o">T</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>27</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve" subscript="o">T</ml:id> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="o">T</ml:id> </ml:apply> <ml:id xml:space="preserve">Θ</ml:id> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="137"/> </region> <region region-id="3565" left="90" top="1802.25" width="97.5" height="45.75" align-x="123" align-y="1836" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="Si">λ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">T</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="300">K</ml:id> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:div/> <ml:real>300</ml:real> <ml:id xml:space="preserve">T</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="138"/> </region> <region region-id="4357" left="132" top="1868.25" width="333.75" height="12" align-x="150" 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="true"> <f family="Arial" charset="0"> <inlineAttr line-through="false">Calculate Temperature versus Time.<sp count="2"/>Find Peak Temp at Given Output</inlineAttr> </f> </text> </region> <region region-id="4333" left="132" top="1898.25" width="79.5" height="12" align-x="165" align-y="1908" 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"> Evaluation Times: </text> </region> <region region-id="4334" left="222" top="1898.25" width="41.25" height="13.5" align-x="234" align-y="1908" 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="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">rt</ml:id> <ml:range> <ml:real>0</ml:real> <ml:real>60</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="139"/> </region> <region region-id="4335" left="282" top="1888.5" width="101.25" height="35.25" align-x="314.25" align-y="1908" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Time</ml:id> <ml:id xml:space="preserve">rt</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>15</ml:real> <ml:id xml:space="preserve" subscript="endμs">time</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>700</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>6</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">rt</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="140"/> </region> <region region-id="4336" left="402" top="1894.5" width="58.5" height="18.75" 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