.MCAD 309000000 \  docDocument MmcObject[T T d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdjj shpRectV  mcDocumentObjectState\ mcPageModelK?DD>J=I?mcHeaderFooterI r.c.maximo@I|P CHeaderFooterJ r.c.maximo@J{n}@J@JMbP?MbP? TextState? TextStyle>@ Times New Roman0,0,0Serial_ParPropDefaultW?Normalfont_style_listO font_styleP  Variables Arial Narrow@P  Constants Arial Narrow@P TextTimes New Roman@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@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@PHeaderRomantic@P FooterArial@P Rotated Math Font Arial Narrow TextRegion* docRegionGshpBoxU c40[[ CharacterMap-RangeMap;@T Passive Lowpass Filter by Network Synthesis LPFSYNTH.mcd ChrPropMap7T RangeElem<T  ChrPropData8 RangeData=~ ParPropMap9T 18.2<38Symbol04<582496<7:@W?18</9<:0@NormalTimes New Roman ;eqRegionB@UA4pX<tree@ p=@@,<>@@@=?@@d>H@@@@p>@A@@@@\w@B@@=@C@@t@B1@D@@@B@E@@t@D1@F@@@D@G@@d@F\e@H@@@F@I@@@@H@J@@d@ITn@K@@p@I@L@@@K\w@M@@@H2@N@B@UhKy]oX@O@@ p@P@@p@O@Q@@@P1@R@B@U(SfeV` @S@@ p@T@@ @S@U@@d@TORIGIN@V@@@T1@W*@UH@7@7-A e being the amplitude of the passband ripple and Tn (w) the Tchebyshev polynomial of order n which, in its transcendental form is represented by . Plotting the amplitude versus frequency response for a network of order 10 and allowing a ripple e = .05 and normalizing for wc = 1 rad/sec.7 @X<@Y8@WSymbol@Z<4@[8@W@X@\<@]8@WSymbol@Z@^<1@_8@W@\@`<*@a8@W@^@b<@c8@W?@`@d<A?8A7SymbolAAB<AC8A7@ Symbol0,0,0A@AD<AE8A7@ Times New Roman0,0,0ABAF<#AG8A7@ Times New Roman0,0,0ADAH<AI8A7@ Times New Roman0,0,0AFAJ<AK8A7@ Times New Roman0,0,0AHAL<AM8A7@ Times New Roman0,0,0AJAN< AO8A7@ Times New Roman0,0,0ALAP<AQ8A7ANAPA:9AR<AS:@W?1AT</AU<AV0@NormalTimes New Roman AW@B@UD5AX@@ pAY@@,AXAZ@@@AYA[@@tAZ1A\@@AZA]@@dA\\eA^@@A\A_@@p@A^A`@@A_Aa@@dA`TnAb@@pA`Ac@@AbAd@@K@AcAe@@Ad1jAf@@AcsAg@@A^2Ah@@AY0Ai@B@UxAj@@ pAk@@pAjAl@@Ak3Am*@Uqii-@Bwith the generalized Tchebyshev polynomial defined by the relation7B9BAn@@dB=nB?@@B=bB@@@B2BA@@@B@BB@@tBA1jBC@@BABD@@dBCsinhBE@@pBCBF@@BEBG@@dBFnBH@@BFaBI@@B@BJ@@dBIsinBK@@pBIBL@@BKBM@@dBLnBN@@BLbBO@@B02BP@@B/BQ@@K@BPBR@@BQ1BS@@BP\eBT@B@UK]XBU@@ pBV@@pBUBW@@BV9BX@B@UyHBY@@ pBZ@@,BYB[@@p@BZB\@@B[B]@@@B\B^@@@B]B_@@dB^coshB`@@pB^Ba@@B`Bb@@dBanBc@@BaaBd@@B]Be@@dBdcosBf@@pBdBg@@BfBh@@dBgnBi@@BgbBj@@B\Bk@@@BjBl@@tBk1jBm@@BkBn@@dBmsinhBo@@pBmBp@@BoBq@@dBpnBr@@BpaBs@@BjBt@@dBssinBu@@pBsBv@@BuBw@@dBvnBx@@BvbBy@@BZBz@@tBy0B{@@ByB|@@tB{1jB}@@{B{B~@@B}\eB@B@UB@@ pB@@pBB@@B10B@B@UJB@@ pB@@,BB@@@BB@@@BB@@dBcoshB@@pBB@@BB@@dBnB@@BaB@@BB@@dBcosB@@pBB@@BB@@dBnB@@BbB@@B0B@B@UpzMB@@ pB@@pBB@@B11B@B@U4QB@@ pB@@,BB@@@BB@@@BB@@tB1jB@@BB@@dBsinhB@@pBB@@BB@@dBnB@@BaB@@BB@@dBsinB@@pBB@@BB@@dBnB@@BbB@@BB@@dB1jB@@{BB@@B\eB@B@UpzRB@@ pB@@pBB@@B12B*@UL;(S<<-@ssince cosh( na ) cannot be equal to zero for any real value of s Eq. (11) can only be satisfied for cos (nb) = 0 7rrsB<B8BB<]B8BBB<B8BBB<B8BSymbolBBB9sB@@CbC@@@C>kCA@B@Uu* CB@@ pCC@@?CBCD@@@CCCE@@,@CDCF@@@CECG@@K@CFCH@@CG1jCI@@CFsCJ@@CECK@@dCJcoshCL@@pCJCM@@CLCN@@dCMaCO@@CMCP@@tCO1jCQ@@COCR@@dCQbCS@@CQkCT@@CDCU@@ @CTCV@@dCUsolveCW@@CUsCX@@CTexpandCY@@CCCZ@B@U+ C[@@ pC\@@pC[C]@@C\15C^*@U -9 Substituting (13) in (15) the 2n roots of Eq ( 3 ) are:7999C_<9C`:@W?1Ca</9Cb<9Cc0@NormalTimes New Roman Cd@B@UCe@@ pCf@@pCeCg@@Cf16Ch@B@UaCi@@ pCj@@,CiCk@@@CjCl@@dCk\lCm@@CkkCn@@CjCo@@@CnCp@@@CoCq@@tCp1jCr@@CpCs@@dCrcoshCt@@pCrCu@@CtaCv@@CoCw@@dCvcosCx@@pCvCy@@CxCz@@@CyC{@@p@CzC|@@C{C}@@@C|C~@@tC}2C@@C}kC@@C|1C@@Cz\pC@@CyC@@tC2C@@CnC@@CnC@@@CC@@dCsinhC@@pCC@@CaC@@CC@@dCsinC@@pCC@@CC@@@CC@@p@CC@@CC@@@CC@@tC2C@@CkC@@C1C@@C\pC@@CC@@tC2C@@CnC*@U,;8 -@SLet us plot those roots in the complex frequency plane using the values in Eq. (1):7S9SC8D.@ Times New Roman0,0,0D;D?<D@8D.@ Times New Roman0,0,0D=DA<DB8D.@ Times New Roman0,0,0D?DC<DD8D.@ Times New Roman0,0,0DADE<DF8D.@ Times New Roman0,0,0DCDG<DH8D.@ Times New Roman0,0,0DEDI<DJ8D.@ Times New Roman0,0,0DGDK<DL8D.@ Times New Roman0,0,0DIDM<DN8D.@ Times New Roman0,0,0DKDO<DP8D.@ Times New Roman0,0,0DMDQ<DR8D.@ Times New Roman0,0,0DODS<DT8D.@ Times New Roman0,0,0DQDU<DV8D.@ Times New Roman0,0,0DSDW<DX8D.@ Times New Roman0,0,0DUDY<DZ8D.@ Times New Roman0,0,0DWD[<D\8D.@ Times New Roman0,0,0DYD]<D^8D.@ Times New Roman0,0,0D[D_<D`8D.@ Times New Roman0,0,0D]Da<Db8D.@ Times New Roman0,0,0D_Dc<Dd8D.@ Times New Roman0,0,0DaDe<Df8D.@ Times New Roman0,0,0DcDg<Dh8D.@ Times New Roman0,0,0DeDi<Dj8D.@ Times New Roman0,0,0DgDk<Dl8D.@ Times New Roman0,0,0DiDm<Dn8D.@ Times New Roman0,0,0DkDo<Dp8D.@ Times New Roman0,0,0DmDq<Dr8D.@ Times New Roman0,0,0DoDs<Dt8D.@ Times New Roman0,0,0DqDu<Dv8D.@ Times New Roman0,0,0DsDw<Dx8D.@ Times New Roman0,0,0DuDy<Dz8D.@ Times New Roman0,0,0DwD{<D|8D.@ Times New Roman0,0,0DyD}<D~8D.@ Times New Roman0,0,0D{D<D8D.@ Times New Roman0,0,0D}D<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD< D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<D8D.@ Times New Roman0,0,0DD<E8D.@ Times New Roman0,0,0DE<E8D.@ Times New Roman0,0,0DE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE <E 8D.@ Times New Roman0,0,0EE <E 8D.@ Times New Roman0,0,0E E <E8D.@ Times New Roman0,0,0E E<E8D.@ Times New Roman0,0,0E E<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E 8D.@ Times New Roman0,0,0EE!<E"8D.@ Times New Roman0,0,0EE#<E$8D.@ Times New Roman0,0,0E!E%<E&8D.@ Times New Roman0,0,0E#E'<E(8D.@ Times New Roman0,0,0E%E)<E*8D.@ Times New Roman0,0,0E'E+<E,8D.@ Times New Roman0,0,0E)E-<E.8D.@ Times New Roman0,0,0E+E/<E08D.@ Times New Roman0,0,0E-E1<E28D.@ Times New Roman0,0,0E/E3<E48D.@ Times New Roman0,0,0E1E5<E68D.@ Times New Roman0,0,0E3E7<E88D.@ Times New Roman0,0,0E5E9<E:8D.@ Times New Roman0,0,0E7E;<E<8D.@ Times New Roman0,0,0E9E=<E>8D.@ Times New Roman0,0,0E;E?<E@8D.@ Times New Roman0,0,0E=EA<EB8D.@ Times New Roman0,0,0E?EC<ED8D.@ Times New Roman0,0,0EAEE<EF8D.@ Times New Roman0,0,0ECEG<EH8D.@ Times New Roman0,0,0EEEI<EJ8D.@ Times New Roman0,0,0EGEK<EL8D.@ Times New Roman0,0,0EIEM<EN8D.@ Times New Roman0,0,0EKEO<EP8D.@ Times New Roman0,0,0EMEQ<ER8D.@ Times New Roman0,0,0EOES<ET8D.@ Times New Roman0,0,0EQEU<EV8D.@ Times New Roman0,0,0ESEW<EX8D.@ Times New Roman0,0,0EUEY<EZ8D.@ Times New Roman0,0,0EWE[<E\8D.@ Times New Roman0,0,0EYE]<E^8D.@ Times New Roman0,0,0E[E_<E`8D.@ Times New Roman0,0,0E]Ea<Eb8D.@ Times New Roman0,0,0E_Ec<Ed8D.@ Times New Roman0,0,0EaEe<Ef8D.@ Times New Roman0,0,0EcEg<Eh8D.@ Times New Roman0,0,0EeEi<Ej8D.@ Times New Roman0,0,0EgEk<El8D.@ Times New Roman0,0,0EiEm<En8D.@ Times New Roman0,0,0EkEo<Ep8D.@ Times New Roman0,0,0EmEq<Er8D.@ Times New Roman0,0,0EoEs<Et8D.@ Times New Roman0,0,0EqEu<Ev8D.@ Times New Roman0,0,0EsEw<Ex8D.@ Times New Roman0,0,0EuEy<Ez8D.@ Times New Roman0,0,0EwE{<E|8D.@ Times New Roman0,0,0EyE}<E~8D.@ Times New Roman0,0,0E{E<E8D.@ Times New Roman0,0,0E}E<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EE<E8D.@ Times New Roman0,0,0EED19E<E:@W?1E</E<E0@NormalTimes New Roman E@B@U( }V '8 E@@ pE@@,EE@@@EE@@dEPE@@pEE@@E\lE@@!EE@@dEmE@@pEE@@EE@@dE\lE@@EE@@dE\lpE@@EmE*@U, > 8 H DD- with pn+1 = 17  E<E8EE<E8E_EE<E8EEEE9 E< E:@W?1E</ E< E0@NormalTimes New Roman E*@U[ z h J -@Since all zeros shall be aligned on the jw axis for a minimum phase network, the expression for the locus of those zeros reduces to: 7E<E8EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EE<E8E@ Times New Roman0,0,0EF<F8E@ Times New Roman0,0,0EF<F8E@ Symbol0,0,0FF<F8E@ Times New Roman0,0,0FF<F8E@ Times New Roman0,0,0FF<F 8E@ Times New Roman0,0,0FF <F 8E@ Times New Roman0,0,0FF <F 8E@ Times New Roman0,0,0F F<F8E@ Times New Roman0,0,0F F<F8E@ Times New Roman0,0,0FF<F8E@ Times New Roman0,0,0FF<F8E@ Times New Roman0,0,0FF<F8E@ Times New Roman0,0,0FF<F8E@ Times New Roman0,0,0FF<F8E@ Times New Roman0,0,0FF<F8E@ Times New Roman0,0,0FF<F8E@ Times New Roman0,0,0FF <F!8E@ Times New Roman0,0,0FF"<F#8E@ Times New Roman0,0,0F F$<F%8E@ Times New Roman0,0,0F"F&<F'8E@ Times New Roman0,0,0F$F(<F)8E@ Times New Roman0,0,0F&F*<F+8E@ Times New Roman0,0,0F(F,<F-8E@ Times New Roman0,0,0F*F.<F/8E@ Times New Roman0,0,0F,F0<F18E@ Times New Roman0,0,0F.F2<F38E@ Times New Roman0,0,0F0F4<F58E@ Times New Roman0,0,0F2F6<F78E@ Times New Roman0,0,0F4F8<F98E@ Times New Roman0,0,0F6F:<F;8E@ Times New Roman0,0,0F8F<<F=8E@ Times New Roman0,0,0F:F><F?8E@ Times New Roman0,0,0FFB<FC8E@ Times New Roman0,0,0F@FD<FE8E@ Times New Roman0,0,0FBFF<FG8E@ Times New Roman0,0,0FDFH<FI8E@ Times New Roman0,0,0FFFJ<FK8E@ Times New Roman0,0,0FHFL<FM8E@ Times New Roman0,0,0FJFN<FO8E@ Times New Roman0,0,0FLFP<FQ8E@ Times New Roman0,0,0FNFR<FS8E@ Times New Roman0,0,0FPFT<FU8E@ Times New Roman0,0,0FRFV<FW8E@ Times New Roman0,0,0FTFX<FY8E@ Times New Roman0,0,0FVFZ<F[8E@ Times New Roman0,0,0FXF\<F]8E@ Times New Roman0,0,0FZF^<F_8E@ Times New Roman0,0,0F\F`<Fa8E@ Times New Roman0,0,0F^Fb<Fc8E@ Times New Roman0,0,0F`Fd<Fe8E@ Times New Roman0,0,0FbFf<Fg8E@ Times New Roman0,0,0FdFh<Fi8E@ Times New Roman0,0,0FfFj<Fk8E@ Times New Roman0,0,0FhFl<Fm8E@ Times New Roman0,0,0FjFn<Fo8E@ Times New Roman0,0,0FlFp<Fq8E@ Times New Roman0,0,0FnFr<Fs8E@ Times New Roman0,0,0FpFt<Fu8E@ Times New Roman0,0,0FrFv<Fw8E@ Times New Roman0,0,0FtFx<Fy8EFvFxE9Fz<F{:@W?1F|</F}<F~0@NormalTimes New Roman F@B@U "  F@@ pF@@,FF@@@FF@@dF\lzF@@FmF@@FF@@@FF@@tF1jF@@FF@@dFcoshF@@pFF@@FaF@@FF@@dFcosF@@pFF@@FF@@@FF@@p@FF@@FF@@@FF@@tF2F@@FkF@@F1F@@F\pF@@FF@@tF2F@@FnF@B@U / "  F@@ pF@@pFF@@F17F*@U I   AA- and equally 7 9 F< F:@W?1F</ F< F0@NormalTimes New Roman F@B@U }6 (  F@@ pF@@,FF@@@FF@@dFQF@@pFF@@F\lF@@!FF@@dFmF@@pFF@@FF@@dF\lF@@FF@@dF\lzF@@FmF*@UA T P  oo-Filter synthesis -7F<F8Fl Times New Roman0,0,09F<F:@W?1F</F<F0@NormalTimes New Roman F*@U\ \z h  LL-@We shall proceed now with a straightforward application of these formulae and conventional methods of filter synthesis for the realization of a given lowpass filter. 79F<F:@W?1F</F<F0@NormalTimes New Roman F*@U  / -"Filter specifications: ( example )7"9"F<"F:@W?1F</"F<"F0@NormalTimes New Roman F@B@U( f V  F@@ pF@@ FF@@dFORIGINF@@F0F@B@U U ! 4 F@@ pF@@ FF@@dFf.cF@@FF@@tF1.8F@@FF@@tF10F@@F8F*@U  5 #{{-cutoff frequency ( Hz )79F<F:@W?1F</F<F0@NormalTimes New Roman F@B@U B ( 6 F@@ pF@@ FF@@dFL.ArF@@KFF@@F.5F*@U  7 #-.maximum passband ripple or attenuation ( dB )7.9.F<.F:@W?1F</.F<.F0@NormalTimes New Roman F@B@U .  8 F@@ pF@@ FF@@dFnF@@F4F*@U  9 #-number of reactive elements79F<F:@W?1F</F<F0@NormalTimes New Roman F@B@U ; % @ F@@ pF@@ FF@@dFR.LF@@F50F*@U   A #on-load resistance ( W )7F<F8FF<F8F@ Symbol0,0,0FF<F8F@ Times New Roman0,0,0FF<G8F@ Times New Roman0,0,0FFF9G<G:@W?1G</G<G0@NormalTimes New Roman G*@U, q; 8 D aa-@W***************************************************************************************7W9WG@@@G=G?@@p@G>G@@@G?GA@@@G@GB@@tGA2GC@@GAmGD@@G@1GE@@G>\pGF@@G=GG@@tGF2GH@@GFnGI@@G2GJ@@@GIGK@@dGJsinhGL@@pGJGM@@GLaGN@@GIGO@@dGNsinGP@@pGNGQ@@GPGR@@@GQGS@@p@GRGT@@GSGU@@@GTGV@@tGU2GW@@GUmGX@@GT1GY@@GR\pGZ@@GQG[@@tGZ2G\@@GZnG]*@U   P  -poles79G^<G_:@W?1G`</Ga<Gb0@NormalTimes New Roman Gc@B@U  # K Gd@@ pGe@@ GdGf@@@GeGg@@dGf\lzGh@@GfmGi@@GeGj@@tGi1jGk@@GiGl@@dGkcosGm@@pGkGn@@GmGo@@@GnGp@@p@GoGq@@GpGr@@@GqGs@@tGr2Gt@@GrmGu@@Gq1Gv@@Go\pGw@@GnGx@@tGw2Gy@@GwnGz*@U  v -zeros79G{<G|:@W?1G}</G~<G0@NormalTimes New Roman G@B@Ux  x G@@ pG@@GG@@@GG@@@GG@@@GG@@tG1.2G@@KGG@@G1.2G@@GG@@@GG@@GG@@ GG@@@GG@@dGImG@@pGG@@GG@@dG\lpG@@GmG@@GG@@dGImG@@pGG@@GG@@dG\lzG@@GmG@@GG@@@GG@@@GG@@tG.1G@@KGG@@G1G@@GG@@@GG@@GG@@ GG@@@GG@@dGReG@@pGG@@GG@@dG\lpG@@GmG@@GG@@dGReG@@pGG@@GG@@dG\lzG@@GmG  )L)L$Poles and Zeros in the complex plane&&&&&&&&&& & & & & &&&G@B@U# 85 0 G@@ pG@@ GG@@dGmG@@GG@@tG0G@@GnG@B@UH v *X l G@@ pG@@ GG@@@GG@@dGPG@@pGG@@G\wG@@!GG@@dGmG@@pGG@@GG@@dG\wG@@GG@@dG\lpG@@GmG@B@UH Cv X G@@ pG@@ GG@@@GG@@dGQG@@pGG@@G\wG@@!GG@@dGmG@@pGG@@GG@@dG\wG@@GG@@dG\lzG@@GmG@B@U a , G@@ pG@@ GG@@@GG@@dG\GG@@pGG@@G\wG@@{GG@@GG@@@GG@@dGQG@@pGG@@G\wG@@GG@@dGPG@@pGG@@G\wG@B@U  G@@ pG@@ GG@@dG\G.maxG@@GG@@tG1G@@GG@@dGcoshG@@pGG@@GG@@dGnG@@GaG@B@U Y G@@ pG@@GG@@@GG@@@GG@@@GG@@vG1G@@G 0.999995G@@GG@@dG\G.maxG@@GG@@GG@@dG\GG@@pGH@@GH@@dH1jH@@H\wH@@GH@@@HH@@@HH@@vH10H@@KHH@@H10H @@HH @@@H H @@H H @@H\wH  )N)^$Reflection Coefficient x Frequency&&&&&&&&& & & & & &&&H@B@U(#b8K0< H@@ pH@@HH@@dH\G.maxH@@HH@@+@HSerial_DisplayNodeXH@@HH*@Ut2 }}-@HWe can derive the input impedance from the input reflection coefficient:7H9HH@@H=\wH?@@H:H@@@dH?QHA@@pH?HB@@HA\wHC@@H9HD@@@HCHE@@dHDPHF@@pHDHG@@HF\wHH@@HCHI@@dHHQHJ@@pHHHK@@HJ\wHL@B@U9C HM@@ pHN@@ HMHO@@@HNHP@@dHONumHQ@@pHOHR@@HQ\wHS@@HNHT@@@HSHU@@dHTPHV@@pHTHW@@HV\wHX@@HSHY@@dHXQHZ@@pHXH[@@HZ\wH\*@U%D ]]-which expands to79H]<H^:@W?1H_</H`<Ha0@NormalTimes New Roman Hb@B@UPE Hc@@ pHd@@HcHe@@,@HdHf@@@HeHg@@dHfNumHh@@pHfHi@@Hh\wHj@@HeHk@@@HjHl@@dHkAHm@@Hk1Hn@@HjHo@@@HnHp@@dHoAHq@@Ho2Hr@@HnHs@@dHr\lHt@@Hr2Hu@@,HdHv@@@HuHw@@@HvHx@@@HwHy@@dHxAHz@@HxnH{@@HwH|@@dH{\lH}@@H{nH~@@HvH@@@H~H@@dHAH@@HH@@dHnH@@H1H@@H~H@@dH\lH@@HH@@dHnH@@H1H@@@HuH@@@HH@@tH1H@@HH@@dHnH@@H1H@@HH@@dHmH@@HH@@@HH@@dHAH@@HmH@@HH@@dH\lH@@HmH@B@U#570@ H@@ pH@@ HH@@@HH@@dHDenH@@pHH@@H\wH@@HH@@@HH@@dHPH@@pHH@@H\wH@@HH@@dHQH@@pHH@@H\wH*@U,%;8A ]]-which expands to79H<H:@W?1H</H<H0@NormalTimes New Roman H@B@UPV8B H@@ pH@@HH@@,@HH@@@HH@@dHDenH@@pHH@@H\wH@@HH@@@HH@@dHBH@@H1H@@HH@@@HH@@dHBH@@H2H@@HH@@dH\lH@@H2H@@,HH@@@HH@@@HH@@@HH@@dHBH@@HnH@@HH@@dH\lH@@HnH@@HH@@@HH@@dHBH@@HH@@dHnH@@H1H@@HH@@dH\lH@@HH@@dHnH@@H1H@@@HH@@@HH@@tH1H@@HH@@dHnH@@H1H@@HH@@dHmH@@HH@@@HH@@dHBH@@HmH@@HH@@dH\lH@@HmH*@U|5 -@aWhere the coefficients of the numerator and denominator polynomials can be calculated as follows:7a9aHIB<IC8I=I@IBI@9@ID<@IE:@W?1IF</@IG<@IH0@NormalTimes New Roman II@B@US `~ IJ@@ pIK@@IJIL@@@IKIM@@dILBIN@@ILmIO@@IKIP@@@IOIQ@@@IPIR@@IPIS@@<@IRIT CVSEqItem EqItem IIIU CVSComboItemCVSItem< q^dCVSOleClientItem ࡱ> Root EntryaB.dK80e@Contents( V ^dj jxArialArialxV<T1c$IV@@IR@XIW@@IOIX@B@UxS` IY@@ pIZ@@IYI[@@@IZI\@@dI[AI]@@I[mI^@@IZI_@@@I^I`@@@I_Ia@@I_Ib@@<@IaIc IXId< zq>dCVSOleClientItem ࡱ> Root EntryaB.dK8@e@Contents( hV >dj jxArialArialxV<T1c$Ie@@Ia@XIf@@I^Ig@B@U[hD Ih@@ pIi@@IhIj@@dIimIk@@IiIl@@@IkIm@@@IlIn@@IlIo@@<@InIp IgIq< qdCVSOleClientItem ࡱ> Root EntryaB.dK8@e@Contents( {V dj jxArialArialxV<T1c$.Ir@@In@XIs@@IkIt@B@Uxj Iu@@ pIv@@,IuIw@@@IvIx@@dIwZ.inIy@@pIwIz@@Iy\wI{@@IvI|@@@I{I}@@@I|I~@@@I}I@@tI~2I@@I~I@@dI\wI@@I5I@@I}I@@tI 2.716866I@@II@@dI\wI@@I3I@@I|I@@tI.504051I@@I\wI@@I{I@@@II@@tI 1.197386I@@II@@dI\wI@@I4I@@II@@tI 1.025455I@@II@@dI\wI@@I2I@B@U0 I@@ pI@@?II@@@II@@@II@@@II@@tI2I@@II@@dI\wI@@I5I@@II@@tI 1.197386I@@II@@dI\wI@@I4I@@II@@dI simplifyI@@ II@@dIfloatI@@I7I@@II*@U Y aa-first element value79I<I:@W?1I</I<I0@NormalTimes New Roman I*@UCbP -@@multiplying the denominator in Zin(w) by the first element value7%#@I<#I8II<I8I@ Symbol0,0,0II<I8I@ Times New Roman0,0,0II<I8IIII9@I<@I:@W?1I</@I<@I0@NormalTimes New Roman I@B@U=g X I@@ pI@@?II@@@II@@@II@@@II@@p@II@@II@@@II@@tI 1.197386I@@II@@dI\wI@@I4I@@II@@tI 1.025455I@@II@@dI\wI@@I2I@@I 1.670305I@@I\wI@@II@@dIexpandI@@ II@@dIfloatI@@I7I@@II*@U| ~~-*subtracting the product from the numerator7*9*I<*I:@W?1I</*I<*I0@NormalTimes New Roman I@B@Uu;, I@@ pI@@?II@@@II@@@II@@p@II@@II@@@II@@@II@@tI2I@@II@@dI\wI@@I5I@@II@@tI 2.716866I@@II@@dI\wI@@I3I@@II@@tI.504051I@@I\wI@@pII@@II@@@II@@tI2.0I@@II@@dI\wI@@I5I@@II@@tI 1.712823I@@II@@dI\wI@@I3J@@IJ@@dJ simplifyJ@@ JJ@@dJfloatJ@@J7J@@IJ*@U -inverted remainder function79J<J:@W?1J </J <J 0@NormalTimes New Roman J @B@UW J @@ pJ@@J J@@@JJ@@@JJ@@tJ 1.197386J@@JJ@@dJ\wJ@@J4J@@JJ@@tJ 1.025455J@@JJ@@dJ\wJ@@J2J@@JJ@@@JJ@@tJ 1.004043J@@JJ@@dJ\wJ@@J3J @@JJ!@@tJ .504051J"@@J \wJ#*@U -)Repeat until the process ends by itself: 7)9)J$<)J%:@W?1J&</)J'<)J(0@NormalTimes New Roman J)@B@U P8 J*@@ pJ+@@?J*J,@@@J+J-@@@J,J.@@@J-J/@@tJ. 1.197386J0@@J.J1@@dJ0\wJ2@@J04J3@@J-J4@@tJ3 1.004043J5@@J3J6@@dJ5\wJ7@@J53J8@@J,J9@@dJ8 simplifyJ:@@ J8J;@@dJ:floatJ<@@J:7J=@@J+J>*@U,;8 ss-second element value79J?<J@:@W?1JA</JB<JC0@NormalTimes New Roman JD@B@Ue?0 JE@@ pJF@@?JEJG@@@JFJH@@@JGJI@@p@JHJJ@@JIJK@@@JJJL@@tJK 1.004043JM@@JKJN@@dJM\wJO@@JM3JP@@JJJQ@@tJP.504051JR@@JP\wJS@@pJHJT@@JSJU@@tJT 1.192564JV@@JT\wJW@@JGJX@@dJWexpandJY@@ JWJZ@@dJYfloatJ[@@JY7J\@@JFJ]@B@U J^@@ pJ_@@?J^J`@@@J_Ja@@@J`Jb@@p@JaJc@@JbJd@@@JcJe@@tJd 1.197386Jf@@JdJg@@dJf\wJh@@Jf4Ji@@JcJj@@tJi 1.025455Jk@@JiJl@@dJk\wJm@@Jk2Jn@@pJaJo@@JnJp@@@JoJq@@tJp 1.197386Jr@@JpJs@@dJr\wJt@@Jr4Ju@@JoJv@@tJu .6011131Jw@@JuJx@@dJw\wJy@@Jw2Jz@@J`J{@@dJz simplifyJ|@@ JzJ}@@dJ|floatJ~@@J|7J@@J_J*@U -inverted remainder function79J<J:@W?1J</J<J0@NormalTimes New Roman J@B@U|B J@@ pJ@@JJ@@@JJ@@@JJ@@tJ 1.004043J@@JJ@@dJ\wJ@@J3J@@JJ@@tJ.504051J@@J\wJ@@JJ@@tJ .4243419J@@JJ@@dJ\wJ@@J2J@B@UIh J@@ pJ@@?JJ@@@JJ@@@JJ@@@JJ@@tJ 1.004043J@@JJ@@dJ\wJ@@J3J@@JJ@@tJ .4243419J@@JJ@@dJ\wJ@@J2J@@JJ@@dJ simplifyJ@@ JJ@@dJfloatJ@@J7J@@JJ*@U\ekh ee-third element value79J<J:@W?1J</J<J0@NormalTimes New Roman J@B@U J@@ pJ@@?JJ@@@JJ@@@JJ@@@JJ@@tJ .4243419J@@JJ@@dJ\wJ@@J2J@@pJJ@@JJ@@tJ 2.366118J@@J\wJ@@JJ@@dJexpandJ@@ JJ@@dJfloatJ@@J7J@@JJ@B@UD5 J@@ pJ@@?JJ@@@JJ@@@JJ@@p@JJ@@JJ@@@JJ@@tJ 1.004043J@@JJ@@dJ\wJ@@J3J@@JJ@@tJ.504051J@@J\wJ@@JJ@@tJ 1.004043J@@JJ@@dJ\wJ@@J3J@@JJ@@dJ simplifyJ@@ JJ@@dJfloatJ@@J7J@@JJ@B@U H8 J@@ pJ@@?JJ@@@JJ@@@JJ@@@JJ@@tJ .4243419J@@JJ@@dJ\wJ@@J2J@@JJ@@tJ.504051J@@J\wJ@@JJ@@dJ simplifyJ@@ JJ@@dJfloatJ@@J7J@@JJ*@U,m;8 mm-fourth element value79J<J:@W?1J</J<J0@NormalTimes New Roman J@B@USJm` J@@ pJ@@ JJ@@@JJ@@dJgJ@@J1J@@J 1.670305K@B@U e K@@ pK@@?KK@@@KK@@@KK@@@KK@@tK.504051K@@K\wK@@pKK @@KK @@tK .841863K @@K \wK @@KK @@dK expandK@@ K K@@dKfloatK@@K7K@@KK@B@UsJ K@@ pK@@ KK@@@KK@@eKFgK@@K2K@@K 1.192564K@B@UJ K@@ pK@@ KK@@@KK@@eKGgK@@K3K@@K 2.366118K @B@U  K!@@ pK"@@?K!K#@@@K"K$@@@K#K%@@@K$K&@@tK% .4243419K'@@K%K(@@dK'\wK)@@K'2K*@@K$K+@@tK* .4243419K,@@K*K-@@dK,\wK.@@K,2K/@@K# simplifyK0@@K"K1@B@UD K2@@ pK3@@ K2K4@@@K3K5@@eK4HgK6@@K44K7@@K3.841863K8*@U:K ^^-B Since this process tends to become very lengthy as the number of elements increases, it was automated by a Mathcad program in the worksheet LPFILT.mcd which is a computational resume of what was explained here. The elements gm are the normalized prototype elements values for w = 1rad/sec and a load resistance of 1 ohm and represent the inductance of a series inductor or the capacitance of a shunt capacitor. We shall proceed, now, by scaling the element values for actual frequency and impedance level in a form adequate for the circuit geometry of the drawing bellow in which the first element, starting from the load resistor, is a series inductor . 7 K9<K:8K8K;< K<8K8mTimes New Roman0,0,0K9K=8K8K;K?<K@8K8_K=KA<3KB8K8K?KC<KD8K8@ Symbol0,0,0KAKE<KF8K8@ Times New Roman0,0,0KCKG<KH8K8@ Times New Roman0,0,0KEKI<KJ8K8@ Times New Roman0,0,0KGKK<KL8K8@ Times New Roman0,0,0KIKM<KN8K8@ Times New Roman0,0,0KKKO<KP8K8@ Times New Roman0,0,0KMKQ<KR8K8@ Times New Roman0,0,0KOKS<KT8K8@ Times New Roman0,0,0KQKU<KV8K8@ Times New Roman0,0,0KSKW<KX8K8@ Times New Roman0,0,0KUKY<KZ8K8@ Times New Roman0,0,0KWK[<K\8K8@ Times New Roman0,0,0KYK]<K^8K8@ Times New Roman0,0,0K[K_<K`8K8@ Times New Roman0,0,0K]Ka<Kb8K8@ Times New Roman0,0,0K_Kc<Kd8K8@ Times New Roman0,0,0KaKe<Kf8K8@ Times New Roman0,0,0KcKg<Kh8K8@ Times New Roman0,0,0KeKi<Kj8K8@ Times New Roman0,0,0KgKk<Kl8K8@ Times New Roman0,0,0KiKm<Kn8K8@ Times New Roman0,0,0KkKo<Kp8K8@ Times New Roman0,0,0KmKq<Kr8K8@ Times New Roman0,0,0KoKs<Kt8K8@ Times New Roman0,0,0KqKu<Kv8K8@ Times New Roman0,0,0KsKw<Kx8K8@ Times New Roman0,0,0KuKy<Kz8K8@ Times New Roman0,0,0KwK{<K|8K8@ Times New Roman0,0,0KyK}<K~8K8@ Times New Roman0,0,0K{K<K8K8@ Times New Roman0,0,0K}K<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<\K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<K8K8@ Times New Roman0,0,0KK<L8K8@ Times New Roman0,0,0KL<L8K8@ Times New Roman0,0,0KL<L8K8@ Times New Roman0,0,0LL<L8K8@ Times New Roman0,0,0LL<L8K8@ Times New Roman0,0,0LL <L 8K8@ Times New Roman0,0,0LL <L 8K8@ Times New Roman0,0,0L L <L8K8@ Times New Roman0,0,0L L<L8K8@ Times New Roman0,0,0L L<L8K8@ Times New Roman0,0,0LL<L8K8@ Times New Roman0,0,0LL<L8K8@ Times New Roman0,0,0LL<L8K8@ Times New Roman0,0,0LL<L8K8@ Times New Roman0,0,0LL<L8K8@ Times New Roman0,0,0LL<L8K8@ Times New Roman0,0,0LL<L 8K8@ Times New Roman0,0,0LL!<L"8K8@ Times New Roman0,0,0LL#<L$8K8@ Times New Roman0,0,0L!L%<L&8K8@ Times New Roman0,0,0L#L'<L(8K8@ Times New Roman0,0,0L%L)<L*8K8@ Times New Roman0,0,0L'L+<L,8K8@ Times New Roman0,0,0L)L-<L.8K8@ Times New Roman0,0,0L+L/<L08K8@ Times New Roman0,0,0L-L1<L28K8@ Times New Roman0,0,0L/L3<L48K8@ Times New Roman0,0,0L1L5<L68K8@ Times New 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MAXIMO2C:\WINDOWS\TEMP\AutoRecovery save of Document1.asd@n@@GTimes New Roman5Symbol3& Arial"1hM"&%Z!0dv  !"#$%&'()*+,-./0123456789:;<=>  FMicrosoft Word Document MSWordDocWord.Document.89q?C bV&WordMicrosoft Word    -@Times New Roman4- . 2  1 .e &@Times New RomaG nbjbjَ &n] P9;;;;;;*;;9999 `j591E.A Guillemin 1951Modern Methods of Network SynthesisAdvances in Electronics L.Marton, Academic Press,Inc N.York E.A.Guillemin 1935Communication Networks Vol: IIN.York, John Wiley & Sons M.E.Van Valkenburg 1982Analog Filter DesignN.York, Holt, Rinehart & Winston O.Brune 1931Synthesis of a finite two-terminal network whose driving-poit impedance is a prescribed function of frequencyJour. of Math. & Phys. vol 10W.Cauer 1958Synthesis of Linear Communications NetworksN.York, Mc Graw-HillW.K.Chen 1986Passive and Active Filters, theory and implementationsN.York John Wiley & Sons xy  lnCJ 5CJH* <Txy0  SlmPD|@$$lF j # $$l j #$y0  Slm  mn$&P/ =!"#$%7SummaryInformation(DocumentSummaryInformation8Ole OlePres000Oh+'04  $,ssssNormalorm1rmMicrosoft Word 8.0@9+@r5Z՜.+,D՜.+,D hp  *************************Dv   Title 6> _PID_GUIDAN{10511E25-A18C-11D1-924B-A74FAFA3DA77}n-*.,2 * E.A Guillemin 1951     ..@2 # Modern Methods of Network Synthesis$  $        .!..2 ! 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