.MCAD 309000000 \  docDocument MmcObject[ d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substance)ǯ?QM: qPQN CCqu'NumericalFormatQdjj shpRectVmcDocumentObjectState\ mcPageModelK????mcHeaderFooterI@I CHeaderFooterJ@J@J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultW?Normalfont_style_listO font_styleP  VariablesTimes New Roman@P  ConstantsTimes New Roman@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@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P BFHeaderArial@P BFFooterArial@P Rotated Math FontTimes New Roman TextRegion* docRegionGshpBoxU XH CharacterMap-RangeMap;!Quartz Crystal Equivalent Circuit ChrPropMap7! RangeElem<!  ChrPropData8 RangeData=}0,0,0 ParPropMap9! <^?:@W,1@@</^@A<^@B0@NormalArial @C*@U#00^((p(p-BDThe impedance between these electrodes is a series resonant circuit (Cs, Ls, Rs) shunted by a parallel capacitor Cp, as shown in an equivalent circuit diagram in reference 4. The network has both a series and parallel resonant frequencies which are typically separated by less than one percent. This equivalent circuit applies to a crystal operating at it's fundamental resonant frequency. Some crystals are specified to operate on a 3rd or 5th overtone in which case one or two additional series resonant arms must be added to the equivalent circuit as stated in reference 4.7D@D*@U H;`HH0H0-ABy definition the parallel (ie anti) resonant frequency is that frequency at which the capacitive and inductive reactances cancel, leaving only the resistive component. This frequency is found by equating the complex term (in the admittance equation above) to zero as follows:7A?<A@8A>0,0,09AA<AB:@W,1AC</AD<AE0@NormalArial AF*@U[0ha@@-@This equation has several solutions. After deleting zero, negative frequency and equivalent solutions (for positive component values ) the following solution remains:7AG<AH8AF0,0,09AI<AJ:@W,1AK</AL<AM0@NormalArial AN@B@U F-pAO@@ pAP@@,AOAQ@@tAP0AR@@APAS@@@ARAT@@K@ASAU@@pATAV@@AUAW@@@AVAX@@dAW\wAY@@AWLsAZ@@AVA[@@tAZ1A\@@pAZA]@@A\A^@@dA]\wA_@@A]CsA`@@pASAa@@A`Ab@@@AaAc@@dAbRsAd@@Ab2Ae@@AaAf@@p@AeAg@@AfAh@@@AgAi@@dAh\wAj@@AhLsAk@@AgAl@@tAk1Am@@pAkAn@@AmAo@@dAn\wAp@@AnCsAq@@Ae2Ar@@ARAs@@dAr\wAt@@ArCpAu@B@UbAv@@ pAw@@,AvAx@@dAw\wpAy@@AwAz@@@AyA{@@@AzA|@@tA{1A}@@A{2A~@@{AzA@@A~2A@@AyA@@{@AA@@AA@@@AA@@@AA@@@AA@@dALsA@@AA@@dACsA@@A2A@@AA@@@AA@@dACpA@@AA@@dARsA@@A2A@@AA@@dACsA@@A2A@@AA@@@AA@@@AA@@tA2A@@ACpA@@ALsA@@ACsA@@AA@@@AA@@dACsA@@pAA@@AA@@tA3A@@A2A@@{AA@@AA@@@AA@@@AA@@@AA@@@AA@@dALsA@@A2A@@ACsA@@AA@@@AA@@@AA@@@AA@@tA2A@@ALsA@@ACsA@@ACpA@@AA@@dARsA@@A2A@@AA@@@AA@@@AA@@dACpA@@A2A@@AA@@dARsA@@A4A@@ACsA@@AA@@@AA@@@AA@@tA4A@@AA@@dACpA@@A2A@@AA@@dARsA@@A2A@@ALsA@@pAA@@AA@@{@AA@@ACpA@@pAA@@AA@@dALsA@@ACsA*@U+(eH-Ignoring Resistance:7A<A8A}0,0,09A<A:@W,1A</A<A0@NormalArial A*@U;7{Hf7/@/@-AWThe Q of these resonance usually lies between 5,000 and 3,000,000 and hence Rs has negligible effect on the resonant frequency and it has been ignored in the following resonant frequency equations. Equating Rs to zero allows the following simplification for network admittance (see reference 3 for Q values near the upper end of this range):7WA@@{B;B?@@B>CsB@@B@UQzhBA@@ pBB@@,BABC@@dBB\wpBD@@BBBE@@{@BDBF@@BEBG@@dBFCsBH@@BFCpBI@@BDBJ@@@BIBK@@{@BJBL@@BKCpBM@@{BJBN@@BMLsBO@@{BIBP@@BOCsBQ*@UsI0II-simplifies to:7BR<BS8BQ0,0,09BT<BU:@W,1BV</BW<BX0@NormalArial BY*@UXX~~-impedance of network7BZ<B[8BY0,0,09B\<B]:@W,1B^</B_<B`0@NormalArial Ba@B@U E/Bb@@ pBc@@,BbBd@@dBcZBe@@BcBf@@tBe1Bg@@BeYBh@B@UBi@@ pBj@@,BiBk@@dBjZBl@@BjBm@@tBl1Bn@@pBlBo@@BnBp@@@BoBq@@K@BpBr@@Bq1jBs@@pBpBt@@BsBu@@@BtBv@@dBu\wBw@@BuLsBx@@BtBy@@tBx1Bz@@pBxB{@@BzB|@@dB{\wB}@@B{CsB~@@BoB@@@B~B@@tB1jB@@B\wB@@B~CpB*@UXX0-by substitution, yields7B<B8B0,0,09B<B:@W,1B</B<B0@NormalArial B@B@U 1/B@@ pB@@,BB@@dBZB@@BB@@K@BB@@B1jB@@pBB@@BB@@@BB@@K@BB@@B1B@@pBB@@BB@@@BB@@dB\wB@@BLsB@@BB@@tB1B@@pBB@@BB@@dB\wB@@BCsB@@BB@@dB\wB@@BCpB*@U f-+result of evaluation over the complex plane7+B<+B8B0,0,09+B<+B:@W,1B</+B<+B0@NormalArial B*@UC8sPj00000-A By definition the series resonant frequency is that frequency at which the capacitive and inductive reactances cancel, leaving only the resistive component. This frequency is found by equating the complex term (in the impedance equation above) to zero as follows:7 B< B8B0,0,09 B< B:@W,1B</ B< B0@NormalArial B*@U=l555-@_has solution (zero and negative frequency solutions, for positive component values, deleted):7_B<_B8B0,0,09_B<_B:@W,1B</_B<_B0@NormalArial B@B@U -B@@ pB@@,BB@@tB0B@@BB@@K@BB@@B1B@@pBB@@BB@@@BB@@K@BB@@B1B@@pBB@@BB@@@BB@@dB\wB@@BLsB@@BB@@tB1B@@pBB@@BB@@dB\wB@@BCsB@@BB@@dB\wB@@BCpB*@Uw} -(series resonant frequency in radians/s. 7(B<(B8B0,0,09(B<(B:@W,1B</(B<(B0@NormalArial B*@U:~JJ- has solution:7 B< B8B0,0,09 B< B:@W,1B</ B< B0@NormalArial B@B@UB@@ pB@@,BB@@dB\wsB@@BB@@tB1B@@pBB@@BB@@{@BB@@BLsB@@{BB@@BCsB*@U E+vU U -ratio of the two frequencies7B<B8B0,0,09B<B:@W,1B</B<B0@NormalArial B@B@U O:(B@@ pB@@,BB@@@BB@@dB\wpC@@B\wsC@@BC@@@CC@@@CC@@tC1C@@CC@@@CC@@{@CC@@CCpC @@{CC @@C LsC @@{CC @@C CsC @@{CC@@C C@@dCCsC@@CCpC@@CC@@tC1C@@pCC@@CC@@{@CC@@CLsC@@{CC@@CCsC@B@U:({C@@ pC@@,CC@@@CC@@dC\wpC@@C\wsC@@CC @@@CC!@@tC 1C"@@{C C#@@C"CpC$@@{CC%@@C$C&@@dC%CsC'@@C%CpC(*@U+9;8II-simplifies to:7C)<C*8C(0,0,09C+<C,:@W,1C-</C.<C/0@NormalArial C0*@U[Skhcc-result of squaring7C1<C28C00,0,09C3<C4:@W,1C5</C6<C70@NormalArial C8@B@URp|C9@@ pC:@@,C9C;@@@C:C<@@@C;C=@@dC<\wpC>@@C<2C?@@C;C@@@dC?\wsCA@@C?2CB@@C:CC@@@CBCD@@dCCCsCE@@CCCpCF@@CBCpCG@B@U ZPxCH@@ pCI@@,CHCJ@@@CICK@@p@CJCL@@CKCM@@dCL\wpCN@@CL\wsCO@@CJ2CP@@CICQ@@p@CPCR@@CQCS@@@CRCT@@tCS1CU@@{CSCV@@CUCpCW@@{CRCX@@CWCY@@dCXCsCZ@@CXCpC[@@CP2C\*@Uk9{xXII-simplifies to:7C]<C^8C\0,0,09C_<C`:@W,1Ca</Cb<Cc0@NormalArial Cd*@UH   -@fResult of dividing each term by Cp and taking square-root. Equivalent to equation 120 in reference 1.7fCe<D?0@NormalArial D@@B@U c 5 DA@@ pDB@@DADC@@dDBCsDD@@DBDE@@+@DD@XDF@@DDpFDG@B@U  @ 50 !DH@@ pDI@@ DHDJ@@dDILsDK@@DIDL@@@DKDM@@@DLDN@@@DMDO@@tDN1.15DP@@DNDQ@@tDP10DR@@DP4DS@@DMDT@@dDShenryDU@@DSmDV@@DLxDW@@DKDX@@dDWyDY@@DWzDZ*@U# 3 0 "??- inductance7 D[< D\8DZ0,0,09 D]< D^:@W,1D_</ D`< Da0@NormalArial Db@B@U $ p5 40 #Dc@@ pDd@@DcDe@@dDdLsDf@@DdDg@@+@Df@XDh@@DfhenryDi@B@U C j ;X $Dj@@ pDk@@ DjDl@@dDk\ws.aDm@@DkDn@@tDm1Do@@pDmDp@@DoDq@@{@DpDr@@DqLsDs@@{DpDt@@DsCsDu*@UK \k X % -0approximate series resonant frequency in radians70Dv<0Dw8Du0,0,090Dx<0Dy:@W,1Dz</0D{<0D|0@NormalArial D}@B@UHL h b` D~@@ pD@@D~D@@dD\ws.aD@@DD@@+@D@XD@@D _n_u_l_l_D*@U{ \ & -&approximate series resonant frequency.7&D<&D8D0,0,09&D<&D:@W,1D</&D<&D0@NormalArial D@B@U w X 7 'D@@ pD@@ DD@@dDfs.aD@@DD@@dD\ws.aD@@DD@@tD2D@@D\pD@B@UH  ^ D@@ pD@@DD@@dDfs.aD@@DD@@+@D@XD@@DkHzD@B@U h 6 (D@@ pD@@ DD@@dDRsD@@DD@@@DD@@dD\ws.aD@@DLsD@@DQD*@U <<- resistance7 D< D8D0,0,09 D< D:@W,1D</ D< D0@NormalArial D@B@UH  ] D@@ pD@@DD@@dDRsD@@DD@@+@D@XD@@Dk\WD@B@U   D@@ pD@@ DD@@dDfpD@@DD@@@DD@@@DD@@tD1D@@DD@@tD4D@@D\pD@@{DD@@D2D@@DD@@{@DD@@DD@@@DD@@@DD@@@DD@@dDLsD@@DD@@dDCsD@@D2D@@DD@@@DD@@dDCpD@@DD@@dDRsD@@D2D@@DD@@dDCsD@@D2D@@DD@@@DD@@@DD@@tD2D@@DCpD@@DLsD@@DCsD@@DD@@@DD@@dDCsD@@pDD@@DD@@tD3D@@D2D@@{DD@@DD@@@DD@@@DD@@@DD@@@DD@@dDLsD@@D2D@@DCsD@@DD@@@DD@@@DD@@@DD@@tD2D@@DLsD@@DCsD@@DCpD@@DD@@dDRsD@@D2D@@DD@@@DD@@@DD@@dDCpD@@D2D@@DD@@dDRsD@@D4D@@DCsD@@DE@@@DE@@@EE@@tE4E@@EE@@dECpE@@E2E@@EE@@dERsE@@E2E @@DLsE @@pDE @@E E @@{@E E @@E CpE@@pE E@@EE@@dELsE@@ECsE@B@U 4 xE 2@ +E@@ pE@@EE@@dEfpE@@EE@@+@E@XE@@EkHzE*@U3 {C @ -"exact parallel resonant frequency.7"E<"E8E0,0,09"E<"E:@W,1E</"E<"E 0@NormalArial E!@B@U3 E @ E"@@ pE#@@ E"E$@@dE#\wpE%@@E#E&@@@E%E'@@dE&fpE(@@E&2E)@@E%\pE*@B@U Q z 8h -E+@@ pE,@@ E+E-@@dE,fp.aE.@@E,E/@@{@E.E0@@E/E1@@dE0CsE2@@E0CpE3@@E.E4@@@E3E5@@@E4E6@@@E5E7@@{@E6E8@@E7CpE9@@{E6E:@@E9LsE;@@{E5E<@@E;CsE=@@E42E>@@E3\pE?*@U[ ,{ h t t -(approximate parallel resonant frequency.7(E@<(EA8E?0,0,09(EB<(EC:@W,1ED</(EE<(EF0@NormalArial EG@B@U\ p h EH@@ pEI@@EHEJ@@dEIfp.aEK@@EIEL@@+@EK@XEM@@EKkHzEN*@U X  -0error in approximate parallel resonant frequency70EO<0EP8EN0,0,090EQ<0ER:@W,1ES</0ET<0EU0@NormalArial EV@B@U j 2 0EW@@ pEX@@ EWEY@@dEXerEZ@@EXE[@@@EZE\@@dE[fp.aE]@@E[fpE^@@EZfpE_@B@U   E`@@ pEa@@E`Eb@@dEaerEc@@EaEd@@+@Ec@XEe@@Ec _n_u_l_l_Ef@B@U f : 1Eg@@ pEh@@EgEi@@@EhEj@@dEifpEk@@Eifs.aEl@@EhEm@@+@El@XEn@@El _n_u_l_l_Eo*@U  2-.ratio of parallel/series resonant frequencies.7.Ep<.Eq8Eo0,0,09.Er<.Es:@W,1Et</.Eu<.Ev0@NormalArial Ew*@U _  3[W0W0-@bandwidth between two frequencies (centered on the series resonant frequency) at which the crystal impedance is 3dB greater than Rs.7Ex<Ey8Ew0,0,09Ez<E{:@W,1E|</E}<E~0@NormalArial E@B@U R 5 4E@@ pE@@ EE@@dE\DfE@@EE@@dEfs.aE@@EQE@B@U 5E@@ pE@@EE@@dE\DfE@@EE@@+@E@XE@@EHzE*@U@# &C @0  -8admittance at the approximate series resonant frequency.78E<8E8E0,0,098E<8E:@W,1E</8E<8E0@NormalArial E@B@U  3` 58 E@@ pE@@ EE@@dEysE@@EE@@p@EE@@EE@@@EE@@tE1E@@pEE@@EE@@dERsE@@EE@@tE1jE@@pEE@@EE@@@EE@@dE\ws.aE@@ELsE@@EE@@tE1E@@pEE@@EE@@dE\ws.aE@@ECsE@@EE@@@EE@@tE1jE@@E\ws.aE@@ECpE@B@U`T m h E@@ pE@@EE@@@EE@@dEysE@@ERsE@@EE@@+@E@XE@@E _n_u_l_l_E*@U@ ; @ 00-@`note that the reactive (ie. imaginary) term above is due to the error in the resonant frequency.7`E<`E8E0,0,09`E<`E:@W,1E</`E<`E0@NormalArial E@B@U ( 6 E@@ pE@@ EE@@dEypE@@EE@@p@EE@@EE@@@EE@@tE1E@@pEE@@EE@@dERsE@@EE@@tE1jE@@pEE@@EE@@@EE@@dE\wpE@@ELsE@@EE@@tE1E@@pEE@@EE@@dE\wpE@@ECsE@@EE@@@EE@@tE1jE@@E\wpE@@ECpE*@U@ H @ -.admittance at the parallel resonant frequency.7.E<.E8E0,0,09.E<.E:@W,1E</.E<.E0@NormalArial E@B@Ux   E@@ pE@@EE@@dEypE@@EE@@+@E@XE@@EsiemensE*@U w  poo-Plot Frequencies:7E<E8E}0,0,0E< E8E}0,0,0EEE9E<E:@W,1E</E<E0@NormalArial E@B@U(+ jR :@ 9E@@ pE@@ EE@@dEf1E@@EF@@dEfs.aF@@EF@@tF1F@@{FF@@F2F*@U3 C @ :QQ-first frequency7F<F8F0,0,09F<F :@W,1F </F <F 0@NormalArial F @B@U(\ am Dh ;F@@ pF@@ FF@@dFspoF@@FF@@tF2F@@FQF*@U[ bk h <-#number of frequencies per octave. 7#F<#F8F0,0,09#F<#F:@W,1F</#F<#F0@NormalArial F*@U^ D h a|[|[-@A large number of plot points is required to cover one octave and accurately show the resonant peaks. For this reason Automatic Mode is turned off. Press [F9] to see graphs.7F<F8F~0,0,09F<F :@W,1F!</F"<F#0@NormalArial F$@B@U(t N > =F%@@ pF&@@ F%F'@@dF&ndF(@@F&1F)*@Us  >ii-number of octaves7F*<F+8F)0,0,09F,<F-:@W,1F.</F/<F00@NormalArial F1@B@Ut _ / /F2@@ pF3@@ F2F4@@dF3m1F5@@F3F6@@dF5spoF7@@F5ndF8@B@U( \ 3 ?F9@@ pF:@@ F9F;@@dF:iF<@@F:F=@@tF<0F>@@F<m1F?*@U C @-range variable for frequencies7F@<FA8F?0,0,09FB<FC:@W,1FD</FE<FF0@NormalArial FG@B@U( l 8 AFH@@ pFI@@ FHFJ@@@FIFK@@dFJfFL@@FJiFM@@FIFN@@dFMf1FO@@FMFP@@tFO2FQ@@FOFR@@dFQiFS@@FQspoFT*@U B``-frequency range7FU<FV8FT0,0,09FW<FX:@W,1FY</FZ<F[0@NormalArial F\@B@U   L !F]@@ pF^@@F]F_@@@F^F`@@dF_fFa@@F_Fb@@dFalastFc@@pFaFd@@FcfFe@@F^Ff@@+@Fe@XFg@@FekHzFh@B@U( c < CFi@@ pFj@@ FiFk@@@FjFl@@dFk\wFm@@FkiFn@@FjFo@@@FnFp@@tFo2Fq@@Fo\pFr@@FnFs@@dFrfFt@@FriFu*@U 1 D-angular velocity, in radians7Fv<Fw8Fu0,0,09Fx<Fy:@W,1Fz</F{<F|0@NormalArial F}@B@U( - 9 EF~@@ pF@@ F~F@@dFYF@@FF@@p@FF@@FF@@@FF@@tF1F@@pFF@@FF@@dFRsF@@FF@@tF1jF@@pFF@@FF@@@FF@@dF\wF@@FLsF@@FF@@tF1F@@pFF@@FF@@dF\wF@@FCsF@@FF@@@FF@@tF1jF@@F\wF@@FCpF*@U    0(kk-crystal admittance7F<F8F0,0,09F<F:@W,1F</F<F0@NormalArial F@B@U` D` F@@ pF@@FF@@@FF@@@FF@@@FF@@BFF@@tF1.978F@@FF@@tF10F@@KFF@@F4F@@FF@@tF1.428F@@FF@@tF10F@@KFF@@F8F@@FF@@dF _n_u_l_l_F@@F _n_u_l_l_F@@FF@@{@FF@@FF@@dFYF@@FiF@@FsiemensF@@FF@@@FF@@@FF@@vF148.413F@@F74.206F@@FF@@dF _n_u_l_l_F@@F _n_u_l_l_F@@FF@@@FF@@dFfF@@FiF@@FkHzF 5 )N)O(Magnitude of crystal admittance verses f&&&&&&&&& & & & & &&&F@B@U`,`F@@ pF@@FF@@@FF@@@FF@@@FF@@vF90F@@KFF@@F88.085F@@FF@@dF _n_u_l_l_F@@F _n_u_l_l_F@@FF@@@FF@@@FF@@dFargF@@pFF@@FF@@dFYF@@FiF@@FF@@tF2F@@F\pF@@F360F@@FF@@@FF@@@FF@@vF148.413F@@F74.206F@@FF@@dF _n_u_l_l_F@@F _n_u_l_l_F@@FF@@@FF@@dFfF@@FiF@@FkHzF 5 )N)N$Angle of crystal admittance verses f&&&&&&&&& & & & & &&&F*@USSc`HKK- References:7 F< F8F}0,0,09 F< F:@W,1F</ F< F0@NormalArial F*@U{<44 4 -@^1. The Royal Signals, Handbook of Line Communications, volume I, 1947, HMSO, pp. 691 - 695.77^F<F8F0,0,0F<F8F{0,0,0FF<'G8F0,0,0FFF9^G<^G:@W,1G</^G<^G0@NormalArial G*@U-@R2. Bliley Electric Co., the Bliley Website, 1988, http://www.bliley.com/index.htm7+RG<G8G0,0,0G <G 8G{0,0,0GG <'G 8G0,0,0G G G9RG