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<editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="1984" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="1539" left="114" top="11.25" width="270.75" height="15.75" align-x="114" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f size="14"> <c val="#000080">Inductance Calulation of Different Shapes</c> </f> </text> </region> <region region-id="1540" left="6" top="38.25" width="466.5" height="96" align-x="6" align-y="48" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">This sheet is used to design planar spiral inductors used on RFICs.<sp count="2"/> </c> <c val="#000080">The expressions are from this paper in the IEEE Journal of Solid State Circuits, 10th October 1999, pages 1419-1424.<sp count="2"/>Paper attached.<sp count="2"/>This paper presents several equations, two are caculated here, the </c> <f family="Arial" charset="0"> <c val="#000080"> <inlineAttr line-through="false">modified Wheeler</inlineAttr> </c> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false"> <c val="#000080"> and a </c> </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false"> <c val="#000080">Current Sheet Approximation</c> </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false"> <c val="#000080">.<sp count="2"/>These </c> </inlineAttr> </f> <c val="#000080">are calculated for the two common planar inductor types and shown with measured results</c> <c val="#000080">The main advantage of this is a starting point for a field solver. </c> </text> </region> <region region-id="1605" left="48" top="156" width="109.5" height="72" align-x="48" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{B801CA65-A1FC-11D0-85AD-444553540000}" type="embedded" item-idref="2"/> <rendering item-idref="3"/> </region> <region region-id="1560" left="222" top="164.25" width="249.75" height="24" align-x="222" align-y="174" 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" size="10"> <inlineAttr line-through="false"> <c val="#000080">Chris Haji-Michael</c> </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr line-through="false"> <c val="#000080">http://www.sunshadow.co.uk/chris.htm</c> </inlineAttr> </f> </text> </region> <region region-id="1894" left="78" top="234" width="331.5" height="144.75" align-x="78" align-y="234" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{D3E34B21-9D75-101A-8C3D-00AA001A1652}" type="embedded" item-idref="4"/> <rendering item-idref="5"/> </region> <region region-id="1603" left="12" top="410.25" width="462" height="24" align-x="12" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">The range of dout is from 100 to 480um; w is from 2 to 0.3*dout; s is 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lock-width="true"> <f family="Arial" charset="0" size="12"> <inlineAttr line-through="false"> <c val="#000080">Approximate Resistance</c> </inlineAttr> </f> <sp/> </text> </region> <region region-id="1882" left="282" top="668.25" width="207" height="48" align-x="282" align-y="678" 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"> <c val="#000080">The resistance is calculated as simply the average circumference x turns x resistivity.<sp count="2"/>This is very crude and ignores corner affects, but will try to allow Q to be plotted..</c> </text> </region> <region region-id="1835" left="6" top="678" width="233.25" height="30" align-x="106.5" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">resistance</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve" subscript="avg">d</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">resistivity</ml:id> </ml:apply> <ml:id xml:space="preserve">w</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="1836" left="276" top="729" width="203.25" height="12.75" align-x="431.25" align-y="738" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">resistance</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>339</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>3.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>10</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">ohm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.92142912529788656</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ohm"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="20"/> </region> <region region-id="1548" left="6" top="774.75" width="180" height="14.25" align-x="6" 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="Arial" charset="0" size="12"> <inlineAttr line-through="false"> <c val="#000080">Square inductor (a) Calculation</c> </inlineAttr> </f> <sp/> </text> </region> <region region-id="1577" left="6" top="813" width="44.25" height="15" align-x="26.25" align-y="822" 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="a1">K</ml:id> <ml:real>2.34</ml:real> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="1578" left="66" top="813" width="44.25" height="15" align-x="86.25" align-y="822" 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="a2">K</ml:id> <ml:real>2.75</ml:real> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="1653" left="156" top="798" width="210" height="38.25" align-x="234.75" align-y="822" 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="smw">L</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="a1">K</ml:id> <ml:id xml:space="preserve" subscript="0">μ</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="avg">d</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="a2">K</ml:id> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="1929" left="96" top="854.25" width="84" height="60" align-x="114" align-y="864" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">Modified wheeler equation for three square inductors compared to measured values</c> <sp/> </text> </region> <region region-id="1930" left="192" top="855" width="178.5" height="15" align-x="332.25" align-y="864" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="smw">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>321</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>3.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>16.5</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>5.75619725375047</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="24"/> </region> <region region-id="1898" left="384" top="854.25" width="93" height="12" align-x="384" align-y="864" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">#10, measured 6.1nH</c> </text> </region> <region region-id="1931" left="192" top="879" width="176.25" height="15" align-x="325.5" align-y="888" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="smw">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>339</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>5.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>10</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>15.329621949263007</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="25"/> </region> <region region-id="1900" left="384" top="878.25" width="98.25" height="12" align-x="384" align-y="888" 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"> <c val="#000080">#15, measured 16.2nH</c> </text> </region> <region region-id="1932" left="192" top="903" width="174" height="15" align-x="332.25" align-y="912" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="smw">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>400</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>3.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>31.6</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>4.7020137564777409</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="26"/> </region> <region region-id="1902" left="384" top="902.25" width="93" height="12" align-x="384" align-y="912" 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"> <c val="#000080">#14, measured 4.9nH</c> </text> </region> <region region-id="1580" left="6" top="945" width="43.5" height="15" align-x="25.5" align-y="954" 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="a1">C</ml:id> <ml:real>1.27</ml:real> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="1581" left="60" top="945" width="43.5" height="15" align-x="79.5" align-y="954" 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="a2">C</ml:id> <ml:real>2.07</ml:real> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="1582" left="126" top="945" width="43.5" height="15" align-x="145.5" align-y="954" 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="a3">C</ml:id> <ml:real>0.18</ml:real> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="1583" left="186" top="945" width="43.5" height="15" align-x="205.5" align-y="954" 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="a4">C</ml:id> <ml:real>0.13</ml:real> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="1728" left="6" top="978" width="462.75" height="38.25" align-x="83.25" align-y="1002" 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="scsa">L</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="0">μ</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="avg">d</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="a1">C</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="a2">C</ml:id> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="a3">C</ml:id> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="a4">C</ml:id> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="1935" left="96" top="1040.25" width="84" height="72" align-x="114" align-y="1050" 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"> <c val="#000080">Current Sheet aproximation for three square inductors compared to measured values</c> <sp/> </text> </region> <region region-id="1916" left="198" top="1041" width="172.5" height="15" align-x="336.75" align-y="1050" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="scsa">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>321</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>3.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>16.5</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>5.69635078200659</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="32"/> </region> <region region-id="1922" left="390" top="1040.25" width="93" height="12" align-x="390" align-y="1050" 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"> <c val="#000080">#10, measured 6.1nH</c> </text> </region> <region region-id="1918" left="198" top="1065" width="170.25" height="15" align-x="330" align-y="1074" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="scsa">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>339</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>5.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>10</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>15.200445747771399</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="33"/> </region> <region region-id="1923" left="390" top="1064.25" width="98.25" height="12" align-x="390" align-y="1074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">#15, measured 16.2nH</c> </text> </region> <region region-id="1920" left="198" top="1089" width="177" height="15" align-x="336.75" align-y="1098" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="scsa">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>400</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>3.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>31.6</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>4.6745571660886061</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="34"/> </region> <region region-id="1924" left="390" top="1088.25" width="93" height="12" align-x="390" align-y="1098" 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"> <c val="#000080">#14, measured 4.9nH</c> </text> </region> <region region-id="1589" left="6" top="1152.75" width="190.5" height="14.25" align-x="6" align-y="1164" 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" size="12"> <inlineAttr line-through="false"> <c val="#000080">Hexagon inductor (c) Calculation</c> </inlineAttr> </f> <sp/> </text> </region> <region region-id="1478" left="6" top="1197" width="44.25" height="15" align-x="26.25" align-y="1206" 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="c1">K</ml:id> <ml:real>2.33</ml:real> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="1479" left="60" top="1197" width="44.25" height="15" align-x="80.25" align-y="1206" 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="c2">K</ml:id> <ml:real>3.82</ml:real> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="1741" left="126" top="1182" width="210.75" height="38.25" align-x="205.5" align-y="1206" 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="hmw">L</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="c1">K</ml:id> <ml:id xml:space="preserve" subscript="0">μ</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="avg">d</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="c2">K</ml:id> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="1956" left="102" top="1226.25" width="84" height="60" align-x="120" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">Modified wheeler equation for two hex inductors compared to measured values</c> <sp/> </text> </region> <region region-id="1944" left="204" top="1239" width="154.5" height="15" align-x="320.25" align-y="1248" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="hmw">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>346</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>4</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>18</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>5.9452431041157912</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="38"/> </region> <region region-id="1937" left="384" top="1238.25" width="93" height="12" align-x="384" align-y="1248" 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"> <c val="#000080">#55, measured 7.5nH</c> </text> </region> <region region-id="1945" left="204" top="1263" width="154.5" height="15" align-x="320.25" align-y="1272" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="hmw">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>326</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>5</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>8</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>12</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>7.2214893418029957</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="39"/> </region> <region region-id="1939" left="384" top="1262.25" width="93" height="12" align-x="384" 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"> <c val="#000080">#58, measured 7.2nH</c> </text> </region> <region region-id="1483" left="6" top="1305" width="43.5" height="15" align-x="25.5" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="c1">C</ml:id> <ml:real>1.09</ml:real> </ml:define> </math> <rendering item-idref="40"/> </region> <region region-id="1484" left="60" top="1305" width="43.5" height="15" align-x="79.5" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="c2">C</ml:id> <ml:real>2.23</ml:real> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="1485" left="126" top="1305" width="39" height="15" align-x="145.5" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="c3">C</ml:id> <ml:real>0.0</ml:real> </ml:define> </math> <rendering item-idref="42"/> </region> <region region-id="1486" left="186" top="1305" width="43.5" height="15" align-x="205.5" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="c4">C</ml:id> <ml:real>0.17</ml:real> </ml:define> </math> <rendering item-idref="43"/> </region> <region region-id="1766" left="6" top="1332" width="463.5" height="38.25" align-x="84" align-y="1356" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="hcsa">L</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="0">μ</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="avg">d</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="c1">C</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="c2">C</ml:id> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="c3">C</ml:id> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="c4">C</ml:id> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="44"/> </region> <region region-id="1959" left="102" top="1382.25" width="84" height="60" align-x="120" align-y="1392" 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"> <c val="#000080">Current Sheet aproximation for two hex inductors compared to measured values</c> <sp/> </text> </region> <region region-id="1948" left="210" top="1395" width="153" height="15" align-x="324.75" align-y="1404" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="hcsa">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>346</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>4</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>18</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>6.0222534078744632</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="45"/> </region> <region region-id="1960" left="384" top="1394.25" width="93" height="12" align-x="384" align-y="1404" 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"> <c val="#000080">#55, measured 7.5nH</c> </text> </region> <region region-id="1949" left="210" top="1419" width="153" height="15" align-x="324.75" align-y="1428" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="hcsa">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>326</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>5</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>8</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>12</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>7.41714710559269</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="46"/> </region> <region region-id="1961" left="384" top="1418.25" width="93" height="12" align-x="384" align-y="1428" 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"> <c val="#000080">#58, measured 7.2nH</c> </text> </region> <region region-id="1980" left="210" top="1524.75" width="138.75" height="14.25" align-x="210" align-y="1536" 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" size="12"> <inlineAttr line-through="false"> <c val="#000080">Plots of inductors</c> </inlineAttr> </f> </text> </region> <region region-id="1861" left="420" top="1527" width="75.75" height="12.75" align-x="450.75" align-y="1536" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Turns</ml:id> <ml:range> <ml:sequence> <ml:real>1</ml:real> <ml:real>1.5</ml:real> </ml:sequence> <ml:real>10</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="1973" left="24" top="1544.25" width="462" height="24" align-x="24" align-y="1554" 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"> <c val="#000080">The inductance increases more with outside diameter presumably because the tracks length is a square of this.</c> </text> </region> <region region-id="1971" left="48" top="1572" width="441" height="179.25" align-x="48" align-y="1572" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="48"/> <rendering item-idref="49"/> </region> <region region-id="1972" left="12" top="1758" width="460.5" height="171" align-x="12" align-y="1758" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="50"/> <rendering item-idref="51"/> </region> <region region-id="1984" left="6" top="1964.25" width="496.5" height="24" align-x="6" align-y="1974" 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"> <c val="#000080">The Q is not correct and should be about 20.<sp count="2"/>This is because the conducted losses are only part of the story and the induced losses into the substrate are more important and not included.</c> </text> </region> <region region-id="1982" left="42" top="2004" width="452.25" height="180" align-x="42" align-y="2004" show-border="false" show-highlight="false" 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customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="1984" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="1539" left="114" top="11.25" width="270.75" height="15.75" align-x="114" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f size="14"> <c val="#000080">Inductance Calulation of Different Shapes</c> </f> </text> </region> <region region-id="1540" left="6" top="38.25" width="466.5" height="96" align-x="6" align-y="48" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">This sheet is used to design planar spiral inductors used on RFICs.<sp count="2"/> </c> <c val="#000080">The expressions are from this paper in the IEEE Journal of Solid State Circuits, 10th October 1999, pages 1419-1424.<sp count="2"/>Paper attached.<sp count="2"/>This paper presents several equations, two are caculated here, the </c> <f family="Arial" charset="0"> <c val="#000080"> <inlineAttr line-through="false">modified Wheeler</inlineAttr> </c> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false"> <c val="#000080"> and a </c> </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false"> <c val="#000080">Current Sheet Approximation</c> </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false"> <c val="#000080">.<sp count="2"/>These </c> </inlineAttr> </f> <c val="#000080">are calculated for the two common planar inductor types and shown with measured results</c> <c val="#000080">The main advantage of this is a starting point for a field solver. </c> </text> </region> <region region-id="1605" left="48" top="156" width="109.5" height="72" align-x="48" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{B801CA65-A1FC-11D0-85AD-444553540000}" type="embedded" item-idref="2"/> <rendering item-idref="3"/> </region> <region region-id="1560" left="222" top="164.25" width="249.75" height="24" align-x="222" align-y="174" 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" size="10"> <inlineAttr line-through="false"> <c val="#000080">Chris Haji-Michael</c> </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr line-through="false"> <c val="#000080">http://www.sunshadow.co.uk/chris.htm</c> </inlineAttr> </f> </text> </region> <region region-id="1894" left="78" top="234" width="331.5" height="144.75" align-x="78" align-y="234" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{D3E34B21-9D75-101A-8C3D-00AA001A1652}" type="embedded" item-idref="4"/> <rendering item-idref="5"/> </region> <region region-id="1603" left="12" top="410.25" width="462" height="24" align-x="12" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">The range of dout is from 100 to 480um; w is from 2 to 0.3*dout; s is from 2um to 3*w; din is from 0.1 to 0.9*dout.<sp count="2"/>Total inductance from 0.5 to 100nH</c> </text> </region> <region region-id="1878" left="384" top="447" width="75" height="18.75" align-x="408" align-y="462" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">freq</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>9</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">Hz</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="1433" left="6" top="452.25" width="44.25" height="33.75" align-x="25.5" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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<ml:real>-7</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">newton</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">amp</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="1663" left="210" top="452.25" width="43.5" height="33.75" align-x="228.75" align-y="468" 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">nH</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">H</ml:id> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>9</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="1821" left="288" top="452.25" width="60.75" height="33.75" align-x="320.25" align-y="468" show-border="false" 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top="507" width="148.5" height="12.75" align-x="363" align-y="516" 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">halfturn</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real font="0">2</ml:real> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>0.25</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="1783" left="6" top="513" width="210" height="15" align-x="76.5" align-y="522" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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left="312" top="543" width="137.25" height="15" align-x="421.5" align-y="552" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="in">d</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>11</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>2.25</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">μm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>3</ml:real> <unitMonomial 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<ml:real>10</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">μm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>170</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="micron"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="16"/> </region> <region region-id="1828" left="6" top="582" width="166.5" height="32.25" align-x="70.5" align-y="600" 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">ρ</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="in">d</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="in">d</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="1829" left="300" top="591" width="159.75" height="12.75" align-x="426" align-y="600" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>321</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>3.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>16.5</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.28760529482551128</ml:real> </result> </ml:eval> </math> <rendering item-idref="18"/> </region> <region region-id="1831" left="6" top="648.75" width="224.25" height="14.25" align-x="6" align-y="660" 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" size="12"> <inlineAttr line-through="false"> <c val="#000080">Approximate Resistance</c> </inlineAttr> </f> <sp/> </text> </region> <region region-id="1882" left="282" top="668.25" width="207" height="48" align-x="282" align-y="678" 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"> <c val="#000080">The resistance is calculated as simply the average circumference x turns x resistivity.<sp count="2"/>This is very crude and ignores corner affects, but will try to allow Q to be plotted..</c> </text> </region> <region region-id="1835" left="6" top="678" width="233.25" height="30" align-x="106.5" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">resistance</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve" subscript="avg">d</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">resistivity</ml:id> </ml:apply> <ml:id xml:space="preserve">w</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="1836" left="276" top="729" width="203.25" height="12.75" align-x="431.25" align-y="738" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">resistance</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>339</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>3.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>10</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">ohm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.92142912529788656</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ohm"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="20"/> </region> <region region-id="1548" left="6" top="774.75" width="180" height="14.25" align-x="6" 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="Arial" charset="0" size="12"> <inlineAttr line-through="false"> <c val="#000080">Square inductor (a) Calculation</c> </inlineAttr> </f> <sp/> </text> </region> <region region-id="1577" left="6" top="813" width="44.25" height="15" align-x="26.25" align-y="822" 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="a1">K</ml:id> <ml:real>2.34</ml:real> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="1578" left="66" top="813" width="44.25" height="15" align-x="86.25" align-y="822" 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="a2">K</ml:id> <ml:real>2.75</ml:real> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="1653" left="156" top="798" width="210" height="38.25" align-x="234.75" align-y="822" 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="smw">L</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="a1">K</ml:id> <ml:id xml:space="preserve" subscript="0">μ</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="avg">d</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="a2">K</ml:id> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="1929" left="96" top="854.25" width="84" height="60" align-x="114" align-y="864" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">Modified wheeler equation for three square inductors compared to measured values</c> <sp/> </text> </region> <region region-id="1930" left="192" top="855" width="178.5" height="15" align-x="332.25" align-y="864" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="smw">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>321</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>3.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>16.5</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>5.75619725375047</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="24"/> </region> <region region-id="1898" left="384" top="854.25" width="93" height="12" align-x="384" align-y="864" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">#10, measured 6.1nH</c> </text> </region> <region region-id="1931" left="192" top="879" width="176.25" height="15" align-x="325.5" align-y="888" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="smw">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>339</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>5.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>10</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>15.329621949263007</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="25"/> </region> <region region-id="1900" left="384" top="878.25" width="98.25" height="12" align-x="384" align-y="888" 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"> <c val="#000080">#15, measured 16.2nH</c> </text> </region> <region region-id="1932" left="192" top="903" width="174" height="15" align-x="332.25" align-y="912" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="smw">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>400</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>3.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>31.6</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>4.7020137564777409</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="26"/> </region> <region region-id="1902" left="384" top="902.25" width="93" height="12" align-x="384" align-y="912" 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"> <c val="#000080">#14, measured 4.9nH</c> </text> </region> <region region-id="1580" left="6" top="945" width="43.5" height="15" align-x="25.5" align-y="954" 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="a1">C</ml:id> <ml:real>1.27</ml:real> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="1581" left="60" top="945" width="43.5" height="15" align-x="79.5" align-y="954" 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="a2">C</ml:id> <ml:real>2.07</ml:real> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="1582" left="126" top="945" width="43.5" height="15" align-x="145.5" align-y="954" 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="a3">C</ml:id> <ml:real>0.18</ml:real> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="1583" left="186" top="945" width="43.5" height="15" align-x="205.5" align-y="954" 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="a4">C</ml:id> <ml:real>0.13</ml:real> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="1728" left="6" top="978" width="462.75" height="38.25" align-x="83.25" align-y="1002" 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="scsa">L</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="0">μ</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="avg">d</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="a1">C</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="a2">C</ml:id> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="a3">C</ml:id> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="a4">C</ml:id> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="1935" left="96" top="1040.25" width="84" height="72" align-x="114" align-y="1050" 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"> <c val="#000080">Current Sheet aproximation for three square inductors compared to measured values</c> <sp/> </text> </region> <region region-id="1916" left="198" top="1041" width="172.5" height="15" align-x="336.75" align-y="1050" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="scsa">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>321</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>3.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>16.5</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>5.69635078200659</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="32"/> </region> <region region-id="1922" left="390" top="1040.25" width="93" height="12" align-x="390" align-y="1050" 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"> <c val="#000080">#10, measured 6.1nH</c> </text> </region> <region region-id="1918" left="198" top="1065" width="170.25" height="15" align-x="330" align-y="1074" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="scsa">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>339</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>5.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>10</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>15.200445747771399</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="33"/> </region> <region region-id="1923" left="390" top="1064.25" width="98.25" height="12" align-x="390" align-y="1074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">#15, measured 16.2nH</c> </text> </region> <region region-id="1920" left="198" top="1089" width="177" height="15" align-x="336.75" align-y="1098" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="scsa">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>400</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>3.75</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>31.6</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1.9</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>4.6745571660886061</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="34"/> </region> <region region-id="1924" left="390" top="1088.25" width="93" height="12" align-x="390" align-y="1098" 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"> <c val="#000080">#14, measured 4.9nH</c> </text> </region> <region region-id="1589" left="6" top="1152.75" width="190.5" height="14.25" align-x="6" align-y="1164" 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" size="12"> <inlineAttr line-through="false"> <c val="#000080">Hexagon inductor (c) Calculation</c> </inlineAttr> </f> <sp/> </text> </region> <region region-id="1478" left="6" top="1197" width="44.25" height="15" align-x="26.25" align-y="1206" 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="c1">K</ml:id> <ml:real>2.33</ml:real> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="1479" left="60" top="1197" width="44.25" height="15" align-x="80.25" align-y="1206" 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="c2">K</ml:id> <ml:real>3.82</ml:real> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="1741" left="126" top="1182" width="210.75" height="38.25" align-x="205.5" align-y="1206" 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="hmw">L</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="c1">K</ml:id> <ml:id xml:space="preserve" subscript="0">μ</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="avg">d</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="c2">K</ml:id> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="1956" left="102" top="1226.25" width="84" height="60" align-x="120" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">Modified wheeler equation for two hex inductors compared to measured values</c> <sp/> </text> </region> <region region-id="1944" left="204" top="1239" width="154.5" height="15" align-x="320.25" align-y="1248" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="hmw">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>346</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>4</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>18</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>5.9452431041157912</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="38"/> </region> <region region-id="1937" left="384" top="1238.25" width="93" height="12" align-x="384" align-y="1248" 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"> <c val="#000080">#55, measured 7.5nH</c> </text> </region> <region region-id="1945" left="204" top="1263" width="154.5" height="15" align-x="320.25" align-y="1272" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="hmw">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>326</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>5</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>8</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>12</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>7.2214893418029957</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="39"/> </region> <region region-id="1939" left="384" top="1262.25" width="93" height="12" align-x="384" 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"> <c val="#000080">#58, measured 7.2nH</c> </text> </region> <region region-id="1483" left="6" top="1305" width="43.5" height="15" align-x="25.5" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="c1">C</ml:id> <ml:real>1.09</ml:real> </ml:define> </math> <rendering item-idref="40"/> </region> <region region-id="1484" left="60" top="1305" width="43.5" height="15" align-x="79.5" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="c2">C</ml:id> <ml:real>2.23</ml:real> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="1485" left="126" top="1305" width="39" height="15" align-x="145.5" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="c3">C</ml:id> <ml:real>0.0</ml:real> </ml:define> </math> <rendering item-idref="42"/> </region> <region region-id="1486" left="186" top="1305" width="43.5" height="15" align-x="205.5" 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:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="c4">C</ml:id> <ml:real>0.17</ml:real> </ml:define> </math> <rendering item-idref="43"/> </region> <region region-id="1766" left="6" top="1332" width="463.5" height="38.25" align-x="84" align-y="1356" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="hcsa">L</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="0">μ</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="avg">d</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="c1">C</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="c2">C</ml:id> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="c3">C</ml:id> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="c4">C</ml:id> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">ρ</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="out">d</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:sequence> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="44"/> </region> <region region-id="1959" left="102" top="1382.25" width="84" height="60" align-x="120" align-y="1392" 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"> <c val="#000080">Current Sheet aproximation for two hex inductors compared to measured values</c> <sp/> </text> </region> <region region-id="1948" left="210" top="1395" width="153" height="15" align-x="324.75" align-y="1404" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="hcsa">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>346</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>4</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>18</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>6.0222534078744632</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="45"/> </region> <region region-id="1960" left="384" top="1394.25" width="93" height="12" align-x="384" align-y="1404" 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"> <c val="#000080">#55, measured 7.5nH</c> </text> </region> <region region-id="1949" left="210" top="1419" width="153" height="15" align-x="324.75" align-y="1428" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="hcsa">L</ml:id> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>326</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:real>5</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:real>8</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>12</ml:real> <ml:id xml:space="preserve">μm</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">nH</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>7.41714710559269</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="2" 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="false" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="46"/> </region> <region region-id="1961" left="384" top="1418.25" width="93" height="12" align-x="384" align-y="1428" 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"> <c val="#000080">#58, measured 7.2nH</c> </text> </region> <region region-id="1980" left="210" top="1524.75" width="138.75" height="14.25" align-x="210" align-y="1536" 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" size="12"> <inlineAttr line-through="false"> <c val="#000080">Plots of inductors</c> </inlineAttr> </f> </text> </region> <region region-id="1861" left="420" top="1527" width="75.75" height="12.75" align-x="450.75" align-y="1536" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Turns</ml:id> <ml:range> <ml:sequence> <ml:real>1</ml:real> <ml:real>1.5</ml:real> </ml:sequence> <ml:real>10</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="1973" left="24" top="1544.25" width="462" height="24" align-x="24" align-y="1554" 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"> <c val="#000080">The inductance increases more with outside diameter presumably because the tracks length is a square of this.</c> </text> </region> <region region-id="1971" left="48" top="1572" width="441" height="179.25" align-x="48" align-y="1572" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="48"/> <rendering item-idref="49"/> </region> <region region-id="1972" left="12" top="1758" width="460.5" height="171" align-x="12" align-y="1758" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="50"/> <rendering item-idref="51"/> </region> <region region-id="1984" left="6" top="1964.25" width="496.5" height="24" align-x="6" align-y="1974" 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"> <c val="#000080">The Q is not correct and should be about 20.<sp count="2"/>This is because the conducted losses are only part of the story and the induced losses into the substrate are more important and not included.</c> </text> </region> <region region-id="1982" left="42" top="2004" width="452.25" 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