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\viewkind4\uc1\pard\f0\fs18\{p\}\par }</left> <center>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fnil\fprq15\fcharset0 Arial;}} \viewkind4\uc1\pard\qc\f0\fs18 Page \{n\} of \{nn\}\par }</center> <right>{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fnil\fprq15\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\f0\fs18 February 2, 2008\par }</right> </footer> </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="true" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC12" local-assignment="MC12"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="true" 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="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="212" 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="2" left="12" top="14.25" width="381" height="15" align-x="12.75" 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="false"> I. Draw the winding function, N FuPUD ( <f family="Symbol" charset="2" size="10"> <inlineAttr font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-style="normal" underline="false" line-through="false">), for a full pitch uniformly distributed coil.</inlineAttr> </f> </text> </region> <region region-id="32" left="30" top="38.25" width="303.75" height="15" align-x="49.5" 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="false"> Determine the equations for NFuPUD(<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">) over the non-horizontal 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</ml:apply> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="1">b</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="4"/> </region> <region region-id="11" left="264" top="122.25" width="12" height="12" align-x="266.25" align-y="132" 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"> => </text> </region> <region region-id="10" left="312" top="116.25" width="42" height="27.75" align-x="327" align-y="132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve" subscript="1">b</ml:id> <ml:apply> <ml:div/> <ml:apply> 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z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:real font="0">3</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="2">b</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="9"/> </region> <region region-id="20" left="264" top="266.25" width="12" height="12" align-x="266.25" align-y="276" 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"> => </text> </region> <region region-id="21" left="312" top="260.25" 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</ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>3</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>3</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>3</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:ifThen> </ml:program> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="33" left="6" top="630" width="407.25" height="306" align-x="6" align-y="630" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="15"/> <rendering item-idref="16"/> </region> <region region-id="30" left="6" top="992.25" width="419.25" height="24" align-x="8.25" align-y="1002" 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"> II.<tab/>Determine the winding series for a full pitch uniformly distributed coil; i.e., determine the Fourier series coefficients of the winding function. <sp/> </text> </region> <region region-id="35" left="18" top="1052.25" width="126" height="12" align-x="37.5" 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="false"> Symmetry Conditions of N(<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)</inlineAttr> </f> </text> </region> <region region-id="44" left="36" top="1082.25" width="188.25" height="15" align-x="37.5" align-y="1092" 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"> a) Even, N(-<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">) = N(</inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)<sp count="4"/>=><sp count="4"/>b</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> h </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> = 0 for all h</inlineAttr> </f> </text> </region> <region region-id="45" left="36" top="1112.25" width="300" height="27" align-x="37.5" 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"> b) Negative half-wave, N(<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a+p</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">) = -N(</inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)<sp count="4"/>=><sp count="4"/>a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> h </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10"> = b</f> </inlineAttr> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> h </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> = 0 for all even h</inlineAttr> </f> </text> </region> <region region-id="46" left="36" 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<ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:program> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:program> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>-4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:program> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:program> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:id xml:space="preserve">α</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:program> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:program> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="21"/> </region> <region region-id="155" left="42" top="1868.25" width="171" height="223.5" align-x="57" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve" subscript="h">a</ml:id> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>-4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>-4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <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">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="22"/> </region> <region region-id="156" left="240" top="1898.25" width="131.25" height="12" align-x="252.75" align-y="1908" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> This term equals zero for all h. </text> </region> <region region-id="157" left="240" top="1982.25" width="131.25" height="12" align-x="252.75" align-y="1992" 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"> This term equals zero for all h. </text> </region> <region region-id="61" left="42" top="2114.25" width="131.25" height="191.25" align-x="57" align-y="2130" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve" subscript="h">a</ml:id> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="23"/> </region> <region region-id="99" left="210" top="2120.25" width="30" height="12" align-x="219.75" align-y="2130" 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"> Term 1 </text> </region> <region region-id="102" left="306" top="2120.25" width="89.25" height="12" align-x="315.75" align-y="2130" 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"> Term 1 + Term 7 = 0 </text> </region> <region region-id="98" left="210" top="2144.25" width="30" height="12" align-x="219.75" align-y="2154" 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"> Term 2 </text> </region> <region region-id="104" left="306" top="2144.25" width="89.25" height="12" align-x="315.75" align-y="2154" 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"> Term 2 + Term 5 = 0 </text> </region> <region region-id="97" left="210" top="2174.25" width="30" height="12" align-x="219.75" align-y="2184" 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"> Term 3 </text> </region> <region region-id="106" left="306" top="2174.25" width="89.25" height="12" align-x="315.75" align-y="2184" 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"> Term 3 + Term 6 = 0 </text> </region> <region region-id="96" left="210" top="2204.25" width="30" height="12" align-x="219.75" align-y="2214" 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"> Term 4 </text> </region> <region region-id="95" left="210" top="2240.25" width="30" height="12" align-x="219.75" align-y="2250" 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"> Term 5 </text> </region> <region region-id="93" left="210" top="2270.25" width="30" height="12" align-x="219.75" align-y="2280" 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"> Term 6 </text> </region> <region region-id="85" left="210" top="2300.25" width="30" height="12" align-x="219.75" align-y="2310" 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"> Term 7 </text> </region> <region region-id="118" left="42" top="2330.25" width="363" height="33.75" align-x="174" align-y="2346" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:equal/> <ml:id xml:space="preserve" subscript="h">a</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </math> <rendering item-idref="24"/> </region> <region region-id="158" left="216" top="2366.25" width="59.25" height="27.75" align-x="262.5" align-y="2382" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="25"/> </region> <region region-id="159" left="294" top="2372.25" width="31.5" height="12" align-x="299.25" align-y="2382" 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"> for all h </text> </region> <region region-id="162" left="48" top="2402.25" width="146.25" height="33.75" align-x="70.5" align-y="2418" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="26"/> </region> <region region-id="161" left="246" top="2402.25" width="153" height="27.75" align-x="268.5" align-y="2418" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="27"/> </region> <region region-id="164" left="246" top="2444.25" width="139.5" height="27.75" align-x="270" align-y="2460" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">a</ml:id> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sinc</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="168" left="18" top="2487" width="53.25" height="12.75" align-x="30" align-y="2496" 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">h</ml:id> <ml:range> <ml:sequence> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:sequence> <ml:real>11</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="134" left="150" top="2484.75" width="124.5" height="32.25" align-x="188.25" align-y="2496" 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="FS">N</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="137" left="6" top="2526" width="407.25" height="246" align-x="6" align-y="2526" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="31"/> <rendering item-idref="32"/> </region> <region region-id="173" left="18" top="2793" width="53.25" height="12.75" align-x="30" align-y="2802" 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">h</ml:id> <ml:range> <ml:sequence> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:sequence> <ml:real>21</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="174" left="108" top="2790.75" width="124.5" height="32.25" align-x="146.25" align-y="2802" 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="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="FS">N</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> </ml:apply> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="166" left="6" top="2838" width="407.25" height="240" align-x="6" align-y="2838" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="35"/> <rendering item-idref="36"/> </region> <region region-id="148" left="12" top="3128.25" width="419.25" height="24" align-x="14.25" align-y="3138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> III.<tab/>Determine the winding factor for a full pitch uniformly distributed coil; i.e., the distribution factor. <sp/> </text> </region> <region region-id="143" left="36" top="3170.25" width="169.5" height="27.75" align-x="90" align-y="3186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="FuPUD">a</ml:id> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sinc</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="150" left="258" top="3176.25" width="168.75" height="12" align-x="263.25" align-y="3186" 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"> for a full pitch uniformly distributed coil </text> </region> <region region-id="177" left="36" top="3212.25" width="115.5" height="27.75" align-x="82.5" align-y="3228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="FuPC">a</ml:id> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="178" left="234" top="3218.25" width="174" height="12" align-x="239.25" align-y="3228" 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"> for a full pitch uniformly distributed coil </text> </region> <region region-id="179" left="36" top="3251.25" width="233.25" height="57.75" align-x="151.5" align-y="3282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve" subscript="FuPUD">k</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve" subscript="FuPUD">a</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="FuPC">a</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sinc</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="39"/> </region> <region region-id="183" left="36" top="3332.25" width="106.5" height="27.75" align-x="90.75" align-y="3348" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="FuPUD">k</ml:id> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">sinc</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> 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</ml:apply> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="1">b</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="4"/> </region> <region region-id="11" left="264" top="122.25" width="12" height="12" align-x="266.25" align-y="132" 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"> => </text> </region> <region region-id="10" left="312" top="116.25" width="42" height="27.75" align-x="327" align-y="132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve" subscript="1">b</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="5"/> </region> <region region-id="16" left="66" top="152.25" width="119.25" height="27.75" align-x="121.5" align-y="168" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve" subscript="FuPUD">N</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="6"/> </region> <region region-id="17" left="30" top="212.25" width="102" height="27.75" align-x="87" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:lessThan/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:real font="0">3</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>3</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="7"/> </region> <region region-id="18" left="66" top="260.25" width="29.25" height="27.75" align-x="78" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve">m</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="8"/> </region> <region region-id="19" left="168" top="260.25" width="66" height="27.75" align-x="177.75" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:real font="0">3</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="2">b</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="9"/> </region> <region region-id="20" left="264" top="266.25" width="12" height="12" align-x="266.25" align-y="276" 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"> => </text> </region> <region region-id="21" left="312" top="260.25" width="54.75" height="27.75" align-x="327" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve" subscript="2">b</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>-3</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="10"/> </region> <region region-id="22" left="66" top="296.25" width="120.75" height="27.75" align-x="121.5" align-y="312" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve" subscript="FuPUD">N</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>3</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="11"/> </region> <region region-id="25" left="66" top="374.25" width="120.75" height="12" align-x="72" align-y="384" 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"> For graphing purposes only: </text> </region> <region region-id="26" left="222" top="375" width="32.25" height="12.75" align-x="235.5" align-y="384" 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">N</ml:id> <ml:real>1.0</ml:real> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="27" left="288" top="368.25" width="31.5" height="27.75" align-x="300" align-y="384" 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">β</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>12</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="23" left="30" top="404.25" width="262.5" height="153.75" align-x="86.25" align-y="420" 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:function> <ml:id xml:space="preserve" subscript="FuPUD">N</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:minus/> <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">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:minus/> <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">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:plus/> <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">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:plus/> <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">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:real font="0">3</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>3</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>3</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>3</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>3</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">N</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:ifThen> </ml:program> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="33" left="6" top="630" width="407.25" height="306" align-x="6" align-y="630" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="15"/> <rendering item-idref="16"/> </region> <region region-id="30" left="6" top="992.25" width="419.25" height="24" align-x="8.25" align-y="1002" 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"> II.<tab/>Determine the winding series for a full pitch uniformly distributed coil; i.e., determine the Fourier series coefficients of the winding function. <sp/> </text> </region> <region region-id="35" left="18" top="1052.25" width="126" height="12" align-x="37.5" 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="false"> Symmetry Conditions of N(<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)</inlineAttr> </f> </text> </region> <region region-id="44" left="36" top="1082.25" width="188.25" height="15" align-x="37.5" align-y="1092" 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"> a) Even, N(-<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">) = N(</inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)<sp count="4"/>=><sp count="4"/>b</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> h </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> = 0 for all h</inlineAttr> </f> </text> </region> <region region-id="45" left="36" top="1112.25" width="300" height="27" align-x="37.5" 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"> b) Negative half-wave, N(<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a+p</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">) = -N(</inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)<sp count="4"/>=><sp count="4"/>a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> h </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10"> = b</f> </inlineAttr> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> h </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> = 0 for all even h</inlineAttr> </f> </text> </region> <region region-id="46" left="36" top="1184.25" width="204.75" height="12" align-x="37.5" align-y="1194" 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"> a & b together =><sp count="2"/>quarter wave symmetry and </text> </region> <region region-id="47" left="270" top="1156.5" width="152.25" height="56.25" align-x="285" align-y="1194" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve" subscript="h">a</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve" subscript="FuPUD">N</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">hα</ml:id> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="17"/> </region> <region region-id="48" left="36" top="1258.5" width="287.25" height="75.75" align-x="51" align-y="1296" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve" subscript="h">a</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:integral 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<ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>-4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <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">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="22"/> </region> <region region-id="156" left="240" top="1898.25" width="131.25" height="12" align-x="252.75" align-y="1908" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> This term equals zero for all h. </text> </region> <region region-id="157" left="240" top="1982.25" width="131.25" height="12" align-x="252.75" align-y="1992" 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"> This term equals zero for all h. </text> </region> <region region-id="61" left="42" top="2114.25" width="131.25" height="191.25" align-x="57" align-y="2130" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:id xml:space="preserve" subscript="h">a</ml:id> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:plus split="true"/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="23"/> </region> <region region-id="99" left="210" top="2120.25" width="30" height="12" align-x="219.75" align-y="2130" 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"> Term 1 </text> </region> <region region-id="102" left="306" top="2120.25" width="89.25" height="12" align-x="315.75" align-y="2130" 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"> Term 1 + Term 7 = 0 </text> </region> <region region-id="98" left="210" top="2144.25" width="30" height="12" align-x="219.75" align-y="2154" 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"> Term 2 </text> </region> <region region-id="104" left="306" top="2144.25" width="89.25" height="12" align-x="315.75" align-y="2154" 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"> Term 2 + Term 5 = 0 </text> </region> <region region-id="97" left="210" top="2174.25" width="30" height="12" align-x="219.75" align-y="2184" 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"> Term 3 </text> </region> <region region-id="106" left="306" top="2174.25" width="89.25" height="12" align-x="315.75" align-y="2184" 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"> Term 3 + Term 6 = 0 </text> </region> <region region-id="96" left="210" top="2204.25" width="30" height="12" align-x="219.75" align-y="2214" 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"> Term 4 </text> </region> <region region-id="95" left="210" top="2240.25" width="30" height="12" align-x="219.75" align-y="2250" 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"> Term 5 </text> </region> <region region-id="93" left="210" top="2270.25" width="30" height="12" align-x="219.75" align-y="2280" 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"> Term 6 </text> </region> <region region-id="85" left="210" top="2300.25" width="30" height="12" align-x="219.75" align-y="2310" 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"> Term 7 </text> </region> <region region-id="118" left="42" top="2330.25" width="363" height="33.75" align-x="174" align-y="2346" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:equal/> <ml:id xml:space="preserve" subscript="h">a</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </math> <rendering item-idref="24"/> </region> <region region-id="158" left="216" top="2366.25" width="59.25" height="27.75" align-x="262.5" align-y="2382" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="25"/> </region> <region region-id="159" left="294" top="2372.25" width="31.5" height="12" align-x="299.25" align-y="2382" 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"> for all h </text> </region> <region region-id="162" left="48" top="2402.25" width="146.25" height="33.75" align-x="70.5" align-y="2418" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="26"/> </region> <region region-id="161" left="246" top="2402.25" width="153" height="27.75" align-x="268.5" align-y="2418" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="27"/> </region> <region region-id="164" left="246" top="2444.25" width="139.5" height="27.75" align-x="270" align-y="2460" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">a</ml:id> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sinc</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="168" left="18" top="2487" width="53.25" height="12.75" align-x="30" align-y="2496" 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">h</ml:id> <ml:range> <ml:sequence> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:sequence> <ml:real>11</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="134" left="150" top="2484.75" width="124.5" height="32.25" align-x="188.25" align-y="2496" 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="FS">N</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="137" left="6" top="2526" width="407.25" height="246" align-x="6" align-y="2526" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="31"/> <rendering item-idref="32"/> </region> <region region-id="173" left="18" top="2793" width="53.25" height="12.75" align-x="30" align-y="2802" 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">h</ml:id> <ml:range> <ml:sequence> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:sequence> <ml:real>21</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="174" left="108" top="2790.75" width="124.5" height="32.25" align-x="146.25" align-y="2802" 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="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="FS">N</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> </ml:apply> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="166" left="6" top="2838" width="407.25" height="240" align-x="6" align-y="2838" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="35"/> <rendering item-idref="36"/> </region> <region region-id="148" left="12" top="3128.25" width="419.25" height="24" align-x="14.25" align-y="3138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> III.<tab/>Determine the winding factor for a full pitch uniformly distributed coil; i.e., the distribution factor. <sp/> </text> </region> <region region-id="143" left="36" top="3170.25" width="169.5" height="27.75" align-x="90" align-y="3186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="FuPUD">a</ml:id> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">h</ml:id> <ml:apply> 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