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<pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="9" orientation="portrait" print-single-page-width="false" page-width="595.5" page-height="842.25"> <margins left="56.63622" right="56.63622" top="42.51969" bottom="42.51969"/> <header use-full-page-width="false"/> <footer use-full-page-width="false"/> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="en" UI="en"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="true" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" 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text-indent="0" text-align="center" list-style-type="none" tabs="inherit"> <f size="14"> <b> <c val="#000080">Accurate Formulas for Calculating Zo of Microstrip with adjustment for line thickness and dispersion</c> </b> </f> </p> </text> </region> <region region-id="2503" left="12" top="56.25" width="447.75" height="48" align-x="12" align-y="66" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="center" list-style-type="none" tabs="inherit"> <c val="#000080">This page calculates Zo for a microstrip line.<sp count="2"/>The formulas are mostly from "Handbook of Microwave and Optical Components", editor K Chang,<sp count="2"/>volume 1, publisher Wiley, table 1.16.<sp count="2"/>There are many equations to calculate Zo but most of them do not properly adjust</c> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#000080"> for line thickness.<sp count="2"/>These results are compared with ADS linecalc results and are excellent. Dispersion is also compensated for.</c> </inlineAttr> </f> </p> </text> </region> <region region-id="2509" left="282" top="110.25" width="194.25" height="24" align-x="282" align-y="120" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="center" list-style-type="none" tabs="inherit"> <f family="Arial" charset="0" size="10"> <b> <inlineAttr line-through="false"> <c val="#000080">Chris Haji-Michael</c> </inlineAttr> </b> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="center" list-style-type="none" tabs="inherit"> <f family="Arial" charset="0" size="10"> <b> <inlineAttr line-through="false"> <c val="#000080">http://www.sunshadow.co.uk/chris.htm</c> </inlineAttr> </b> </f> </p> </text> </region> <region region-id="2358" left="12" top="140.25" width="56.25" height="12" align-x="27.75" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0"> <b> <inlineAttr line-through="false"> <c val="#000080">Definitions: </c> </inlineAttr> </b> </f> </p> </text> </region> <region region-id="2111" left="12" top="159" width="58.5" height="18.75" align-x="31.5" align-y="174" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">μm</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">m</ml:id> <ml:apply> <ml:pow/> 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show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="true"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">u</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="2497" left="204" top="158.25" width="33" height="27.75" align-x="216" align-y="174" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="true"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">g</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="2498" left="252" top="158.25" width="33.75" height="27.75" align-x="264.75" align-y="174" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="true"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">p</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="1887" left="12" top="200.25" width="183.75" height="12" align-x="36" align-y="210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0"> <b> <inlineAttr line-through="false"> <c val="#000080">Compensate for conductor thickness (t)</c> </inlineAttr> </b> </f> </p> </text> </region> <region region-id="2466" left="18" top="224.25" width="240.75" height="90.75" align-x="60.75" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="eff">u</ml:id> <ml:boundVars> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:id xml:space="preserve">u</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">u</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>1.25</ml:real> <ml:parens> <ml:apply> <ml:div/> <ml:id 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<ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:real>12</ml:real> <ml:id xml:space="preserve">u</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>-0.5</ml:real> </ml:apply> <ml:apply> <ml:mult/> <ml:real>0.04</ml:real> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:id xml:space="preserve">u</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>4.6</ml:real> </ml:apply> <ml:id xml:space="preserve">p</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:sqrt/> <ml:id xml:space="preserve">u</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:greaterThan/> <ml:id xml:space="preserve">u</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:div/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:real>12</ml:real> <ml:id xml:space="preserve">u</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>-0.5</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>4.6</ml:real> </ml:apply> <ml:id xml:space="preserve">p</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:sqrt/> <ml:id xml:space="preserve">u</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:ifThen> </ml:program> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="2465" left="12" top="470.25" width="426.75" height="102" align-x="66" align-y="486" 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">Zo</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:apply> <ml:id xml:space="preserve" subscript="eff">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>60</ml:real> <ml:apply> <ml:sqrt/> <ml:apply> <ml:id xml:space="preserve" subscript="eff">ε</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>8</ml:real> <ml:apply> <ml:id xml:space="preserve" subscript="eff">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:real>0.25</ml:real> <ml:apply> <ml:id xml:space="preserve" subscript="eff">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve" subscript="eff">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>60</ml:real> <ml:apply> <ml:sqrt/> <ml:apply> <ml:id xml:space="preserve" subscript="eff">ε</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>8</ml:real> <ml:apply> <ml:id xml:space="preserve" subscript="eff">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:real>0.25</ml:real> <ml:apply> <ml:id xml:space="preserve" subscript="eff">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:greaterThan/> <ml:apply> <ml:id xml:space="preserve" subscript="eff">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>120</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:id xml:space="preserve" subscript="eff">ε</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve" subscript="eff">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> <ml:real>1.393</ml:real> </ml:apply> <ml:apply> <ml:mult/> <ml:real>0.667</ml:real> <ml:apply> <ml:id xml:space="preserve">ln</ml:id> <ml:apply> <ml:plus/> <ml:real>1.444</ml:real> <ml:apply> <ml:id xml:space="preserve" subscript="eff">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ohm</ml:id> </ml:apply> </ml:ifThen> </ml:program> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="2264" left="6" top="596.25" width="49.5" height="12" align-x="23.25" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0"> <b> <inlineAttr line-through="false"> <c val="#000080">Dispersion</c> </inlineAttr> </b> </f> </p> </text> </region> <region region-id="2267" left="6" top="614.25" width="467.25" height="24" align-x="14.25" align-y="624" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="center" list-style-type="inherit" tabs="inherit"> <c val="#000080">The effective permitivity is influenced by the maximum frequency limit of the microstrip.<sp count="2"/>This little-known adjustment is from Pramanick & Bhartia</c> </p> </text> </region> <region region-id="2468" left="330" top="644.25" width="144.75" height="60" align-x="330" align-y="654" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="center" list-style-type="none" tabs="inherit"> <c val="#000080">The permittivity is plotted for typical IC dimensions against frequency.<sp count="2"/>It can be seen that dispersion need only be considered above 1E12 Hz</c> </p> </text> </region> <region region-id="2469" left="12" top="652.5" width="309.75" height="31.5" align-x="92.25" align-y="672" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">F</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> <ml:id xml:space="preserve">freq</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:parens> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:id xml:space="preserve">freq</ml:id> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:real>0.5</ml:real> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:apply> <ml:id xml:space="preserve">log</ml:id> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:id xml:space="preserve" subscript="eff">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="2471" left="12" top="698.25" width="184.5" height="27.75" align-x="130.5" align-y="714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">F</ml:id> <ml:sequence> <ml:real>4.1</ml:real> <ml:apply> <ml:div/> <ml:real>10</ml:real> <ml:real>7</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:real>7</ml:real> </ml:apply> <ml:apply> <ml:mult/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>9</ml:real> </ml:apply> <ml:id xml:space="preserve">Hz</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>7</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>0.00066477299324318464</ml:real> </result> </ml:eval> </math> <rendering item-idref="11"/> </region> <region region-id="2474" left="12" top="784.5" width="301.5" height="45.75" align-x="103.5" align-y="810" 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="eff.f">ε</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> <ml:id xml:space="preserve">freq</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:sqrt/> <ml:id xml:space="preserve" subscript="r">ε</ml:id> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:id xml:space="preserve" subscript="eff">ε</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">F</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> <ml:id xml:space="preserve">freq</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:real>-1.5</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:id xml:space="preserve" subscript="eff">ε</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="2476" left="12" top="856.5" width="360.75" height="33.75" align-x="102" align-y="876" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="o.f">Z</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> <ml:id xml:space="preserve">freq</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">Zo</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> <ml:parens> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="eff.f">ε</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> <ml:id xml:space="preserve">freq</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="eff">ε</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve" subscript="eff">ε</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="eff.f">ε</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="r">ε</ml:id> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">p</ml:id> <ml:id xml:space="preserve">freq</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="2477" left="12" top="912" width="431.25" height="177.75" align-x="12" align-y="912" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="14"/> <rendering item-idref="15"/> </region> <region region-id="2489" left="18" top="1112.25" width="444" height="12" align-x="25.5" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="center" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0"> <b> <inlineAttr line-through="false"> <c val="#000080">The value of Zo compares well with values calculated by HPADS and APLAC.</c> </inlineAttr> </b> </f> </p> </text> </region> <region region-id="2481" left="108" top="1136.25" width="110.25" height="27.75" align-x="170.25" align-y="1152" 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">Zo</ml:id> <ml:sequence> <ml:real>4.1</ml:real> <ml:apply> <ml:div/> <ml:real>5</ml:real> <ml:real>7</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:real>7</ml:real> </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>75.2601129482231</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ohm"/> </unitMonomial> 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