.MCAD 309000000 \  docDocument MmcObject[99 d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdjj shpRectVXmcDocumentObjectState\ mcPageModelK].?>].?>mcHeaderFooterI@I |P CHeaderFooterJA"{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq14 Arial;}{\f3\fswiss\fprq15 Arial;}{\f4\fscript\fprq2 EngrvrsOldEng Bd BT;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f4\fs16 roald c. maximo\plain\f3\fs18 \par } A{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq14 Arial;}{\f3\fswiss\fprq15 Arial;}{\f4\fscript\fprq2 EngrvrsOldEng Bd BT;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f3\fs18 \par } A{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq14 Arial;}{\f3\fswiss\fprq15 Arial;}{\f4\fscript\fprq2 EngrvrsOldEng Bd BT;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \par } @JA{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq14 Arial;}{\f3\fswiss\fprq15 Arial;}{\f4\fscript\fprq2 EngrvrsOldEng Bd BT;}{\f5\fswiss Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f3\fs18 \par } A{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq14 Arial;}{\f3\fswiss\fprq15 Arial;}{\f4\fscript\fprq2 EngrvrsOldEng Bd BT;}{\f5\fswiss Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f5\fs20 \par } A{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fnil\fprq14 Arial;}{\f3\fswiss\fprq15 Arial;}{\f4\fscript\fprq2 EngrvrsOldEng Bd BT;}{\f5\fswiss Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f3\fs18 \{n\} \par } @J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultWNormal>@Arial@W Heading 1>@ Arial@W Heading 2 >@ Arial@W Heading 3 >@ Arial@W Paragraph >@ Arial@WList >@ Arial@WIndent >@Times New Roman@W Title>@Times New Roman@W Subtitle font_style_listO font_styleP VariablesMS Serif@P ConstantsMS Serif@P TextArial@P Greek VariablesSymbol@P User 1Arial@P User 2 Courier New@P User 3System@P User 4Script@P User 5Roman@P User 6Modern@P User 7Times New Roman@PSymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@PRotated Math FontMS Serif TextRegion* docRegionGshpBoxUK2 $ CharacterMap-RangeMap;-Roald C. Maximo, Nov. 2000 roaldmax@ig.com.br ChrPropMap7- RangeElem< ChrPropData8 RangeData=l Arial<8n Arial ParPropMap9-<- ParPropData:@WEmbedMap1<LinkMap/-<-LinkData0@NormalArial *@U,(- The Synthesis of Active Filters7 < 8lArial9 < :@W1</ < 0@NormalArial !*@UKkX  -AThe purpose of this worksheet is just to illustrate the use of Mathcad in a step by step synthesis method as applied to active filter design. As as an example, a fifth order active lowpass filter incorporating a third and a second order sections in cascade will be implemented.79"<#:@W$<%:@W"$"1&</'<(0@NormalArial )*@U{/  -@Referring, first, to the circuit in Fig ( 1 ) we can see that its transfer function can be expressed in terms of the feed-forward and feedback transfer functions of the network by the equations:793*<+:@W,<3-:@W*,,1.<//<00@NormalArial 1*@U` ``-Feed forward transfer function792<3:@W14</5<60@NormalArial 7eqRegionB@U0P8tree@ p9@@,8:@@@9;@@d:H.ff<@@p:=@@<s>@@9?@@@>@@@@@?@A@@d@@V.3@B@@p@@@C@@@Bs@D@@?@E@@d@DV.1@F@@p@D@G@@@Fs@H@@,>@I@@@@H@J@@d@IV.2@K@@p@I@L@@@Ks@M@@@H0@N*@U*-(1)7@O<@P8@NmArial@Q<@R8@N@O@S<@T8@NmArial@Q@U<@V8@N@S@U@O9@W<@X:@W1@Y</@Z<@[0@NormalArial @\*@U`#3`0h-Feedback transfer function79@]<@^:@W1@_</@`<@a0@NormalArial @b@B@U0TR0@c@@ p@d@@,@c@e@@@@d@f@@d@eH.fb@g@@p@e@h@@@gs@i@@@d@j@@@@i@k@@@@j@l@@d@kV.3@m@@p@k@n@@@ms@o@@@j@p@@d@oV.2@q@@p@o@r@@@qs@s@@,@i@t@@@@s@u@@d@tV.1@v@@p@t@w@@@vs@x@@@s0@y*@U#30+-(2)7@z<@{8@y@|<@}8@ymArial@z@~<@8@y@|@~@|9@<@:@W1@</@<@0@NormalArial @ COleDocItemDocItem @Upepttࡱ> *  !"#$%&'(),-./012345678Root EntryFzTCompObjqVisioDocument JVisioInformation" FMicrosoft Visio DrawingVISIO 4.0 ShapesVisio.Drawing.69qOh+'08DP\ht1$Visio (TM) Drawing  JRKH2d !fffMMM333bq0bT d Arial-3Times New Roman?-":5T?? 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R TPU !"#%&'()*+T{-./J125U6789:;=>?@2BCTFUUU U UUUUU !"+Ul4,#t@@6iK? uC-_bA,Ul4, dA-\37"A_*<N@4p PRVgfq4u i5H<( H<( _*<NE|[ RVgLfhq{ '* g"4FXh(W@(_arGy 16:TR>$7 UTu&_\8zJ_O{J(CO߯c$L d|ơN, )14-$=2 V@`?;F=C>B@@@C=C?@@@C>C@@@tC?2CA@@C?CB@@dCACCC@@CA1CD@@C>CE@@dCDRCF@@CD2CG@@C=CH@@dCGCCI@@CG3CJ@@COCx<Cy:@WCz<C{:@WCxC|<>C}:@WCzC|C|1C~</OC@@@D=D?@@@D>D@@@@D?DA@@@D@DB@@@DADC@@@DBDD@@@DCDE@@@DDDF@@dDEsDG@@DE3DH@@DDDI@@dDHCDJ@@DH1DK@@DCDL@@dDKCDM@@DK2DN@@DBDO@@dDNRDP@@DN3DQ@@DADR@@dDQCDS@@DQ3DT@@D@DU@@p@DTDV@@DUDW@@@DVDX@@@DWDY@@@DXDZ@@tDY2D[@@DYD\@@dD[CD]@@D[1D^@@DXD_@@dD^RD`@@D^2Da@@DWDb@@dDaCDc@@Da3Dd@@DVDe@@@DdDf@@@DeDg@@tDf2Dh@@DfDi@@dDhCDj@@Dh1Dk@@DeDl@@dDkCDm@@Dk2Dn@@DdDo@@dDnRDp@@Dn2Dq@@DTDr@@dDqsDs@@Dq2Dt@@D?Du@@p@DtDv@@DuDw@@@DvDx@@dDwRDy@@DwDz@@dDyCD{@@Dy3D|@@DvD}@@@D|D~@@tD}3D@@D}RD@@D|D@@dDCD@@D1D@@DtsD@@D>1D@@ D=D@@dDcoeffsD@@DsD@@D<D*@U   -( 11 )7D<D8DmArial9D<D:@W1D</D<D0@NormalArial D*@U h  ]``-?The coefficients of the denominator polynomial in ( 10.a ) are:7?9?DD@@ pD@@ DD@@dDf.cD@@DD@@tD2.1D@@DD@@tD10D@@D3D*@U %#-<== Cutoff frequency ( Hz )79D<D:@W?1D</D<D0@NormalArial D@B@U-%<8D@@ pD@@ DD@@dDripD@@D1D*@U+%;8#-<== Passband ripple ( dB )79D<D:@W?1D</D<D0@NormalArial D@B@UM$\XD@@ pD@@ DD@@dDND@@D5D*@UK[X.^^-<== Filter order79D<D:@W1D</D<D0@NormalArial D*@U(s(-@S***********************************************************************************7S9SD -( 15 )7E0<E18E/mArial9E2<E3:@W1E4</E5<E60@NormalArial E7@B@U7E8@@ pE9@@ E8E:@@dE9mE;@@E9E<@@tE;1E=@@E;NE>@B@U6E?@@ pE@@@ E?EA@@dE@\iEB@@E@EC@@tEB1ED@@EB3EE@B@U5EF@@ pEG@@ EFEH@@dEG\kEI@@EGEJ@@tEI4EK@@EI5EL@B@U5  EM@@ pEN@@ EMEO@@@ENEP@@dEO\lpEQ@@EOmER@@ENES@@@ERET@@K@ESEU@@ETEV@@dEUsinhEW@@pEUEX@@EWaEY@@ESEZ@@dEYsinE[@@pEYE\@@E[E]@@@E\E^@@dE]\pE_@@pE]E`@@E_Ea@@@E`Eb@@tEa2Ec@@EamEd@@E`1Ee@@E\Ef@@tEe2Eg@@EeNEh@@EREi@@@EhEj@@tEi1jEk@@EiEl@@dEkcoshEm@@pEkEn@@EmaEo@@EhEp@@dEocosEq@@pEoEr@@EqEs@@@ErEt@@dEs\pEu@@pEsEv@@EuEw@@@EvEx@@tEw2Ey@@EwmEz@@Ev1E{@@ErE|@@tE{2E}@@E{NE~*@UbK&&-poles7E<E8E~~ 9E<E:@W?1E</E<E0@NormalArial E*@Uc-( 16 )7E<E8EmArial9E<E:@W1E</E<E0@NormalArial E@B@U00E@@ pE@@EE@@@EE@@@EE@@@EE@@tE2E@@KEE@@E2E@@EE@@dE _n_u_l_l_E@@E _n_u_l_l_E@@EE@@dEImE@@pEE@@EE@@dE\lpE@@EmE@@EE@@@EE@@@EE@@tE.5E@@KEE@@E1E@@EE@@dE _n_u_l_l_E@@E _n_u_l_l_E@@EE@@dEReE@@pEE@@EE@@dE\lpE@@EmE  )L)LPoles in the complex plane&&&&&&&&&& & & & & &&&E*@USP`'HH4H4-@Here we chose the three inner poles lp2, lp3 and lp4 to form the 3rd. order Hurwitz polynomial (P3) and the two outer poles lp1, lp5 for the 2nd. order one (P2).7$E<$E8EE<E8E@ SymbolEE<E8E@ ArialEE<E8E@ ArialEE<E8E@ ArialEE<E8E@ SymbolEE<E8E@ ArialEE<E8E@ ArialEE<E8E@ ArialEE<E8E@ SymbolEE<E8E@ ArialEE<E8E@ ArialEE<-E8E@ ArialEE<E8E@ ArialEE<E8E@ ArialEE<E8E}SymbolEE<E8E}EE<E8E]EE<E8E]EE<E8E}SymbolEE<E8E}EE<E8E]EE<E8E@ ArialEE<E8E@ ArialEE<E8E@ ArialEEE9E<E:@W?1E</E<E0@NormalArial E@B@U}(E@@ pE@@ EE@@@EE@@dEP.3E@@pEE@@E\wE@@AEE@@@EE@@tE2E@@E4E@@EE@@dEmE@@pEE@@EE@@dE\wE@@EE@@dE\lpE@@EmE*@U-( 17 )7E<E8EmArial9E<E:@W1F</F<F0@NormalArial F@B@U(F@@ pF@@ FF@@@FF@@dFP.2F@@pFF @@F\wF @@FF @@p@F F @@F F @@dF \wF@@F F@@dF\lpF@@F1F@@pF F@@FF@@dF\wF@@FF@@dF\lpF@@F5F*@U""-( 17a)7F<F8FmArial9F<F:@W1F</F<F0@NormalArial F*@U;KH-!homogenizing to conform with (11)7!9!F @@dF= CoeffP.3F?@@F=1F@@B@UhETvPFA@@ pFB@@FAFC@@dFBAFD@@FBFE@@+@FDSerial_DisplayNodeXFF@@FD5T`FG@B@U0yFH@@ pFI@@ FHFJ@@dFI CoeffP.2FK@@?FIFL@@@FKFM@@@FLFN@@dFMP.2FO@@pFMFP@@FO\wFQ@@FLFR@@ @FQFS@@dFRcoeffsFT@@FR\wFU@@ FQFV@@dFUfloatFW@@FU5FX@@FKFY*@U|-!homogenizing to conform with (12)7!9!FZ@B@U,G?@@ pG@@@ G?GA@@@G@GB@@dGACGC@@GA5GD@@G@GE@@@GDGF@@dGEBGG@@GE3GH@@GDGI@@dGHCGJ@@GH4GK*@U-)And the five normalized capacitances are:7)9)GL<)GM:@W1GN</)GO<)GP0@NormalArial GQ@B@U5*D@GR@@ pGS@@GRGT@@dGSCGU@@GSGV@@+@GU@XGW@@GU4J0`GX*@Up;KpHN-@B<== normalized filter capacitance's for wc = 1 Hz and R's = 1 Ohm.71(BGY<(GZ8GXG[<G\8GX@ SymbolGYG]<G^8GXG[G_<G`8GXArialG]G_G[9BGa%8H*@@ pH+@@ H*H,@@@H+H-@@dH,HH.@@pH,H/@@H.\wH0@@H+H1@@@H0H2@@dH1H.3H3@@pH1H4@@H3\wH5@@H0H6@@dH5H.2H7@@pH5H8@@H7\wH9*@U+;83-( 18 )7H:<H;8H9mArial9H<<H=:@W1H></H?<H@0@NormalArial HA@B@U `/ `HB@@ pHC@@HBHD@@@HCHE@@@HDHF@@@HEHG@@tHF0HH@@KHFHI@@HH30HJ@@HEHK@@tHJ1HL@@KHJHM@@HLripHN@@HDHO@@tHN20HP@@HNHQ@@dHPlogHR@@pHPHS@@{HRHT@@HSHU@@dHTHHV@@pHTHW@@HVHX@@tHW1jHY@@HW\wHZ@@HCH[@@@HZH\@@@H[H]@@@H\H^@@tH]3.5H_@@H]H`@@tH_10Ha@@H_3Hb@@H\0Hc@@H[Hd@@PHcHe@@Hcf.cHf@@HZHg@@dHf\wHh@@Hff.cHi + )\)\$Overall filter transf. function (dB)&&&&&&&&& & & & & &&&Hj*@Uk {x-@ZScaling the components for the specified cutoff frequency and practical resistance values:7Z9ZHk Root EntryaB.dK8y@Contents( {V dj jxArialArialxV<T1c$H@@H@XH@@HH@B@UH}cTH@@ pH@@HH@@@HH@@dHC.actH@@HmH@@HH@@@HH@@@HH@@HH@@<@HH HH< JLdCVSOleClientItem ࡱ> Root EntryaB.dK8y@Contents( V Ldj jtxArialArialxV<T1c$H@@H@XH@@H2J0aH*@U-+<== Actual circuit capacitances in Farads7+9+H<+H:@W1H</+H<+H0@NormalArial H*@U^6h.|.|-AReferences: Roald.C. Maximo, www.mathsoft.com/appsindex.html/LPFSYNTH.mcd L.A. Pipes, The Matrix Theory of Four Terminal Networks Phil.Mag. ser.7-33 November, 1940 P.I. Richards, Applications of Matrix Algebra to Filter Theory, Proc. I.R.E. March, 1946 M.E. Fisher, The Matrix Approach to Filters and Transmission Lines, Electronic Eng. May, June, 1955 Wai-Kai Chen, Passive and Active Filters Theory and Implementations, John Wiley & Sons, New York 19867 H< H8HnTimes New RomanH<H8HlTimes New RomanHH<.H8HnTimes New RomanHH<H8HHH< H8HlTimes New RomanHH<.H8HnTimes New RomanHH<H8HjTimes New RomanHH<I8HnTimes New RomanHI< I8HlTimes New RomanHI<3I8HnTimes New RomanII< I8HjTimes New RomanII< I8HnTimes New RomanII < I 8HlTimes New RomanII <9I 8HnTimes New RomanI I <I8HjTimes New RomanI I<I8HnTimes New RomanI I< I8HlTimes New RomanII<9I8HnTimes New RomanII<I8HjTimes New RomanII<I8HnTimes New RomanIIH9JJ|I