<description/> <author>chris</author> <company/> <keywords/> <revisedBy>chris</revisedBy> </userData> <identityInfo> <revision>2</revision> <documentID>FC6547F9-C64F-4027-A675-B347C06CC425</documentID> <versionID>E3D37E31-9EE2-4362-893F-20FE449D66C8</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> </textStyles> </textRendering> <mathRendering equation-color="#000"> <operators multiplication="dot" derivative="derivative" literal-subscript="small" definition="colon-equal" 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align-y="30" 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="14"> <inlineAttr line-through="false"> <c val="#000080">Microwave Notch Filter Design Sheet</c> </inlineAttr> </f> </text> </region> <region region-id="749" left="288" top="20.25" width="169.5" height="24" align-x="288" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0"> <inlineAttr line-through="false"> <c val="#000080">Chris Haji-Michael www.sunshadow.co.uk/chris.htm</c> </inlineAttr> </f> </text> </region> <region region-id="767" left="12" top="56.25" width="438" height="24" align-x="20.25" 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"> <c val="#000080">This sheet calculates a stop-band filter according to Schiffman and Matthaei, IEEE trans MTT, January 1962. page 6.</c> </text> </region> <region region-id="715" left="48" top="114" width="358.5" height="260.25" align-x="48" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <png item-idref="1" display-width="357" display-height="258.75"/> </picture> <rendering item-idref="2"/> </region> <region region-id="738" left="108" top="126.75" width="22.5" height="14.25" align-x="108" align-y="138" 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">N=3</c> </inlineAttr> </f> </text> 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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">There are three steps to this notch filter:</c> <f family="Arial" charset="0"> <inlineAttr line-through="false"> <c val="#000080">Step 1</c> </inlineAttr> </f> <c val="#000080">, Get the filter g-values, this sheet calculated the g values according to the Chebychev polynomial for a given N, but these can be obtained in a variety of ways from books or from a program freely available from this web page.</c> <f family="Arial" charset="0"> <inlineAttr line-through="false"> <c val="#000080">Step 2</c> </inlineAttr> </f> <c val="#000080">, Calculate Zo for the filter (this sheet stops there).<sp count="2"/>All section are lambda/4</c> <c val="#000080"> <inlineAttr line-through="false"> <f family="Arial" charset="0">Step 3</f> </inlineAttr> </c> <c val="#000080">, Calculate length and width.<sp count="2"/>One method is to use Linecalc which is part of ADS.<sp count="2"/>For microstrip designs I have written a MathCAD sheet which is available from my wepage that works well for thin tracks, less well for thicker tracks. Also you can try </c> <inlineAttr line-through="false"> <c val="#000080"> <f family="Arial" charset="0">transcalc.sourceforge.net</f> </c> </inlineAttr> <c val="#000080"> for a Linecalc equivalent.</c> <c val="#000080">Yellow is user input, Green is output</c> </text> </region> <region region-id="744" left="282" top="897" width="64.5" height="18.75" align-x="301.5" align-y="912" 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:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-6</ml:real> </ml:apply> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="747" left="372" top="897" width="80.25" height="18.75" align-x="390.75" align-y="912" 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:mult/> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-9</ml:real> </ml:apply> <ml:id xml:space="preserve">henry</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="480" left="6" top="912.75" width="195.75" height="14.25" align-x="6" align-y="924" 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">Main user input area:</c> </inlineAttr> </f> </text> </region> <region region-id="285" left="12" top="939" width="48" height="15" align-x="40.5" align-y="948" 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" subscript="ar_db">L</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="374" left="90" top="938.25" width="223.5" height="12" align-x="90" align-y="948" 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">Passband ripple in dB for the chebchev calculations</c> </text> </region> <region region-id="149" left="12" top="963" width="27" height="12.75" align-x="26.25" align-y="972" show-border="false" show-highlight="true" 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">N</ml:id> <ml:real>5</ml:real> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="150" left="90" top="962.25" width="326.25" height="12" align-x="90" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <c val="#000080">Order of the filter 1 to 5, please choose correct section below for the results</c> </text> </region> <region region-id="1082" left="294" top="1004.25" width="162.75" height="36" align-x="294" align-y="1014" 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">refer to frequency repetition above.<sp count="2"/>If you need to attenuate signal and all harmonics set to 1.5 </c> </text> </region> 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<ml:real>2</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="421" left="12" top="1130.25" width="110.25" height="33.75" align-x="30" align-y="1152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">b</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">γ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">N</ml:id> </ml:apply> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="422" left="156" top="1143" width="30" height="18" align-x="173.25" align-y="1152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="314" left="222" top="1148.25" width="227.25" height="36" align-x="222" align-y="1158" 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">g(0) and g(N+1) represent the input/output coupling</c> <c val="#000080">for odd order filters, these are 1 representing the generator (and equal) load resistance</c> </text> </region> <region region-id="423" left="156" top="1167" width="29.25" height="18" align-x="172.5" align-y="1176" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="424" left="144" top="1191" width="42" height="18" align-x="173.25" align-y="1200" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">N</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="807" left="12" top="1215" width="135" height="75" align-x="29.25" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:program> <ml:ifThen> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:real>2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:ifThen> <ml:otherwise> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:real>4</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:otherwise> </ml:program> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="808" left="372" top="1201.5" width="57" height="111.75" align-x="382.5" align-y="1260" 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:id xml:space="preserve">g</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="7" cols="1"> <ml:real>1</ml:real> <ml:real>1.1468378278006446</ml:real> <ml:real>1.3712099875101429</ml:real> <ml:real>1.975027577855714</ml:real> <ml:real>1.3712099875101427</ml:real> <ml:real>1.1468378278006448</ml:real> <ml:real>1</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="810" left="6" top="1314.75" width="129.75" height="14.25" align-x="6" align-y="1326" 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">Calculate basic factors</c> </inlineAttr> </f> </text> </region> <region region-id="830" left="12" top="1344" width="57.75" height="30" align-x="26.25" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="0">f</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="bw">f</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="832" left="102" top="1344" width="66.75" height="32.25" align-x="114" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">a</ml:id> <ml:apply> <ml:id xml:space="preserve">cot</ml:id> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:id xml:space="preserve" subscript="0">f</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="846" left="198" top="1353" width="42.75" height="12.75" align-x="212.25" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Λ</ml:id> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve">ωp</ml:id> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="837" left="378" top="1353" width="61.5" height="15" align-x="390.75" align-y="1362" 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:id xml:space="preserve" subscript="1">f</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">GHz</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.12</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="gigahertz"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="26"/> </region> <region region-id="842" left="294" top="1389" width="57.75" height="12.75" align-x="304.5" align-y="1398" 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:id xml:space="preserve">a</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.50952544949442891</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="6" 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="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="27"/> </region> <region 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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">For N=1.<sp count="2"/>Note the input and output impedances are different</c> </inlineAttr> </f> </text> </region> <region region-id="688" left="12" top="1470" width="78" height="35.25" align-x="45.75" align-y="1488" 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="1">N1_Z</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="A">Z</ml:id> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve">Λ</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="752" left="120" top="1467" width="70.5" height="38.25" align-x="155.25" align-y="1488" 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="B">N1_Z</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve" subscript="A">Z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="865" left="354" top="1491" width="90.75" height="15" align-x="386.25" align-y="1500" 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:id xml:space="preserve" subscript="1">N1_Z</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">ohm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>85.566174132438533</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ohm"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="31"/> </region> <region region-id="866" left="354" top="1515" width="87.75" height="15" align-x="387.75" align-y="1524" 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:id xml:space="preserve" subscript="B">N1_Z</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">ohm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>68.560499375507149</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ohm"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="32"/> </region> <region region-id="756" left="6" top="1560.75" width="45" height="14.25" align-x="6" align-y="1572" 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">For N=2</c> </inlineAttr> </f> 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<ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="889" left="354" top="1653" width="90.75" height="15" align-x="386.25" align-y="1662" 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:id xml:space="preserve" subscript="2">N2_Z</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">ohm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>71.564914323183956</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ohm"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="39"/> </region> <region region-id="890" left="354" top="1677" width="92.25" 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align-y="30" 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="14"> <inlineAttr line-through="false"> <c val="#000080">Microwave Notch Filter Design Sheet</c> </inlineAttr> </f> </text> </region> <region region-id="749" left="288" top="20.25" width="169.5" height="24" align-x="288" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0"> <inlineAttr line-through="false"> <c val="#000080">Chris Haji-Michael www.sunshadow.co.uk/chris.htm</c> </inlineAttr> </f> </text> </region> <region region-id="767" left="12" top="56.25" width="438" height="24" align-x="20.25" 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"> <c val="#000080">This sheet calculates a stop-band filter according to Schiffman and Matthaei, IEEE trans MTT, January 1962. page 6.</c> </text> </region> <region region-id="715" left="48" top="114" width="358.5" height="260.25" align-x="48" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <png item-idref="1" display-width="357" display-height="258.75"/> </picture> <rendering item-idref="2"/> </region> <region region-id="738" left="108" top="126.75" width="22.5" height="14.25" align-x="108" align-y="138" 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">N=3</c> </inlineAttr> </f> </text> 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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">There are three steps to this notch filter:</c> <f family="Arial" charset="0"> <inlineAttr line-through="false"> <c val="#000080">Step 1</c> </inlineAttr> </f> <c val="#000080">, Get the filter g-values, this sheet calculated the g values according to the Chebychev polynomial for a given N, but these can be obtained in a variety of ways from books or from a program freely available from this web page.</c> <f family="Arial" charset="0"> <inlineAttr line-through="false"> <c val="#000080">Step 2</c> </inlineAttr> </f> <c val="#000080">, Calculate Zo for the filter (this sheet stops there).<sp count="2"/>All section are lambda/4</c> <c val="#000080"> <inlineAttr line-through="false"> <f family="Arial" charset="0">Step 3</f> </inlineAttr> </c> <c val="#000080">, Calculate length and width.<sp count="2"/>One method is to use Linecalc which is part of ADS.<sp count="2"/>For microstrip designs I have written a MathCAD sheet which is available from my wepage that works well for thin tracks, less well for thicker tracks. Also you can try </c> <inlineAttr line-through="false"> <c val="#000080"> <f family="Arial" charset="0">transcalc.sourceforge.net</f> </c> </inlineAttr> <c val="#000080"> for a Linecalc equivalent.</c> <c val="#000080">Yellow is user input, Green is output</c> </text> </region> <region region-id="744" left="282" top="897" width="64.5" height="18.75" align-x="301.5" align-y="912" 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:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-6</ml:real> </ml:apply> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="747" left="372" top="897" width="80.25" height="18.75" align-x="390.75" align-y="912" 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:mult/> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-9</ml:real> </ml:apply> <ml:id xml:space="preserve">henry</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="480" left="6" top="912.75" width="195.75" height="14.25" align-x="6" align-y="924" 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">Main user input area:</c> </inlineAttr> </f> </text> </region> <region region-id="285" left="12" top="939" width="48" height="15" align-x="40.5" align-y="948" 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" subscript="ar_db">L</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="374" left="90" top="938.25" width="223.5" height="12" align-x="90" align-y="948" 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">Passband ripple in dB for the chebchev calculations</c> </text> </region> <region region-id="149" left="12" top="963" width="27" height="12.75" align-x="26.25" align-y="972" show-border="false" show-highlight="true" 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">N</ml:id> <ml:real>5</ml:real> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="150" left="90" top="962.25" width="326.25" height="12" align-x="90" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <c val="#000080">Order of the filter 1 to 5, please choose correct section below for the results</c> </text> </region> <region region-id="1082" left="294" top="1004.25" width="162.75" height="36" align-x="294" align-y="1014" 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">refer to frequency repetition above.<sp count="2"/>If you need to attenuate signal and all harmonics set to 1.5 </c> </text> </region> <region region-id="151" left="12" top="1011" width="59.25" height="15" align-x="26.25" align-y="1020" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="0">f</ml:id> <ml:apply> <ml:mult/> <ml:real>1.6</ml:real> <ml:id xml:space="preserve">GHz</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="805" left="84" top="1011" width="55.5" height="15" align-x="103.5" align-y="1020" show-border="false" show-highlight="true" 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="bw">f</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="0">f</ml:id> <ml:real>0.6</ml:real> </ml:apply> 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item-idref="12"/> </region> <region region-id="418" left="6" top="1048.5" width="216.75" height="17.25" align-x="6" align-y="1062" 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">Calculate Chebychev (g) Polynomials</c> </inlineAttr> </f> <f family="Comic Sans MS" charset="0" size="12"> <inlineAttr line-through="false"> <c val="#000080"> <sp/> </c> </inlineAttr> </f> </text> </region> <region region-id="419" left="12" top="1089" width="39.75" height="12.75" align-x="24" align-y="1098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>1</ml:real> <ml:id 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z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000080">g(0) and g(N+1) represent the input/output coupling</c> <c val="#000080">for odd order filters, these are 1 representing the generator (and equal) load resistance</c> </text> </region> <region region-id="423" left="156" top="1167" width="29.25" height="18" align-x="172.5" align-y="1176" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="424" left="144" top="1191" width="42" height="18" align-x="173.25" align-y="1200" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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<ml:real>2</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:ifThen> <ml:otherwise> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:real>4</ml:real> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">b</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:otherwise> </ml:program> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="808" left="372" top="1201.5" width="57" height="111.75" align-x="382.5" align-y="1260" 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:id xml:space="preserve">g</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="7" cols="1"> <ml:real>1</ml:real> <ml:real>1.1468378278006446</ml:real> <ml:real>1.3712099875101429</ml:real> <ml:real>1.975027577855714</ml:real> <ml:real>1.3712099875101427</ml:real> <ml:real>1.1468378278006448</ml:real> <ml:real>1</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="810" left="6" top="1314.75" width="129.75" height="14.25" align-x="6" align-y="1326" 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">Calculate basic factors</c> </inlineAttr> </f> </text> </region> <region region-id="830" left="12" top="1344" width="57.75" height="30" align-x="26.25" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="0">f</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="bw">f</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="832" left="102" top="1344" width="66.75" height="32.25" align-x="114" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">a</ml:id> <ml:apply> <ml:id xml:space="preserve">cot</ml:id> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:id xml:space="preserve" subscript="0">f</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="846" left="198" top="1353" width="42.75" height="12.75" align-x="212.25" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Λ</ml:id> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve">ωp</ml:id> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="837" left="378" top="1353" width="61.5" height="15" align-x="390.75" align-y="1362" 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:id xml:space="preserve" subscript="1">f</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">GHz</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.12</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="gigahertz"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="26"/> </region> <region region-id="842" left="294" top="1389" width="57.75" height="12.75" align-x="304.5" align-y="1398" 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:id xml:space="preserve">a</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.50952544949442891</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="6" 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="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="27"/> </region> <region 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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">For N=1.<sp count="2"/>Note the input and output impedances are different</c> </inlineAttr> </f> </text> </region> <region region-id="688" left="12" top="1470" width="78" height="35.25" align-x="45.75" align-y="1488" 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="1">N1_Z</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="A">Z</ml:id> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve">Λ</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="752" left="120" top="1467" width="70.5" height="38.25" align-x="155.25" align-y="1488" 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="B">N1_Z</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve" subscript="A">Z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="865" left="354" top="1491" width="90.75" height="15" align-x="386.25" align-y="1500" 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:id xml:space="preserve" subscript="1">N1_Z</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">ohm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>85.566174132438533</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ohm"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="31"/> </region> <region region-id="866" left="354" top="1515" width="87.75" height="15" align-x="387.75" align-y="1524" 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:id xml:space="preserve" subscript="B">N1_Z</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">ohm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>68.560499375507149</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ohm"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="32"/> </region> <region region-id="756" left="6" top="1560.75" width="45" height="14.25" align-x="6" align-y="1572" 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">For N=2</c> </inlineAttr> </f> </text> </region> <region region-id="757" left="12" top="1592.25" width="125.25" height="33" align-x="45.75" align-y="1608" 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="1">N2_Z</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="A">Z</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve">Λ</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="879" left="180" top="1599" width="123" height="18" align-x="217.5" align-y="1608" 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="12">N2_Z</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="A">Z</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve">Λ</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="867" left="354" top="1605" width="95.25" height="15" align-x="386.25" align-y="1614" 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:id xml:space="preserve" subscript="1">N2_Z</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">ohm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>135.56617413243853</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ohm"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="35"/> </region> <region region-id="884" left="354" top="1629" width="94.5" height="15" align-x="390" align-y="1638" 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:id xml:space="preserve" subscript="12">N2_Z</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">ohm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>79.2171529853669</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ohm"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="36"/> </region> <region region-id="885" left="12" top="1641" width="69" height="38.25" align-x="45.75" align-y="1662" 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="2">N2_Z</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve" subscript="A">Z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve">Λ</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="886" left="144" top="1653" width="80.25" height="18" align-x="179.25" align-y="1662" 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="B">N2_Z</ml:id> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve" subscript="A">Z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="889" left="354" top="1653" width="90.75" height="15" align-x="386.25" align-y="1662" 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:id xml:space="preserve" subscript="2">N2_Z</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">ohm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>71.564914323183956</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ohm"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="39"/> </region> <region region-id="890" left="354" top="1677" width="92.25" height="15" align-x="387.75" align-y="1686" 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:id xml:space="preserve" subscript="B">N2_Z</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">ohm</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>98.7513788927857</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="ohm"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="40"/> </region> <region region-id="909" left="6" top="1716.75" width="45" height="14.25" align-x="6" align-y="1728" 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">For N=3</c> </inlineAttr> </f> </text> </region> <region region-id="910" left="12" top="1748.25" width="125.25" height="33" align-x="45.75" align-y="1764" 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="1">N3_Z</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="A">Z</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve">Λ</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="911" left="180" top="1755" width="123" height="18" align-x="217.5" align-y="1764" 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="12">N3_Z</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="A">Z</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="narrow-dot"/> <ml:apply> <ml:mult style="narrow-dot"/> <ml:id xml:space="preserve">Λ</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> 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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">Does it work?</c> </inlineAttr> </f> </text> </region> <region region-id="1178" left="18" top="2574" width="435.75" height="240.75" align-x="18" align-y="2574" 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="90"/> <rendering item-idref="91"/> </region> <region region-id="1180" left="18" top="2840.25" width="423.75" height="36" align-x="18" align-y="2850" 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 plots above are for four different notch filterss from 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