.MCAD 306000000 \  docDocumentMmcObject[qq d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecurrent temperature luminosity substanceNumericalFormatQdii shpRectVH+!-R!mcDocumentObjectState\ mcPageModelK????mcHeaderFooterI@I |P ComputeEngineNBuiltInsS SerialAnyvalY@@Y @@Y@YMbP?@Y units_classR TextState? TextStyle>@ Times New RomanNormal>@Times New Roman Heading 1>@ Arial Heading 2 >@ Arial Heading 3 >@ Arial Paragraph >@ ArialList >@ ArialIndent >@Times New RomanTitle>@Times New RomanSubtitle font_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 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Romank TextRegion* docRegionGshpBoxU0$ 9HH CharacterMap-RangeMap;Example  ChrPropMap7 ParPropMap9 RangeElem< ParPropData: RangeData=EmbedMap1<LinkMap/<LinkData0@NormalTimes New Roman *@U1]D!@;`MM- Define Units:7 9 < :1</ < 0@NormalTimes New Roman eqRegionB@Up1F@@tree@ p@@ @@dkVA@@kW @B@U1$F@A!@@ p"@@ !#@@d"kVAR$@@"kW%*@UXhq  : :-@A 3-f load draws 200kW at a PF of 0.707 lagging from a 440V line. In parallel is a 3-f capacitor bank that supplies 50kVAR. Find the resultant power factor and current (magnitude) into the parallel combination.7V&<' ChrPropData8%n (<)8%@ Symbol&*@@ =?@@d>PF@@@@>0.707@A@B@U0XM@B@@ p@C@@ @B@D@@d@C\f.PF@E@@@C@F@@d@Eacos@G@@p@E@H@@@G0.707@I@B@UGO@J@@ p@K@@@J@L@@d@K\f.PF@M@@@K@N@@+@@MSerial_DisplayNodeX@O@@@Mdeg@P@B@U(YJ@Q@@ p@R@@ @Q@S@@d@RV.Line@T@@5@R@U@@t@T440@V@@@TV@W@B@U0bN@X@@ p@Y@@ @X@Z@@d@YQ.Cap@[@@5@Y@\@@t@[50@]@@@[kVAR@^@B@U(3X K@_@@ p@`@@ @_@a@@d@`Sload@b@@@`@c@@d@bP.Load@d@@@bPF@e@B@U& @f@@ p@g@@@f@h@@d@gSload@i@@@g@j@@+@@i@X@k@@@ikVA@l@B@U`HdX@m@@ p@n@@ @m@o@@d@nQload@p@@@n@q@@d@pSload@r@@@p@s@@d@rsin@t@@p@r@u@@@t\f.PF@v@B@UI^IXP@w@@ p@x@@@w@y@@d@xQload@z@@@x@{@@+@@z@X@|@@@zkVAR@}*@U@yF0YY-Or in one step:79@~<@:1@</@<@0@NormalTimes New Roman @@B@U`h@@@ p@@@ @@@@d@Q.Load@@@@@@@d@P.Load@@@@@@@d@tan@@@p@@@@@\f.PF@@B@U@w+@@@ p@@@@@@@d@Q.Load@@@@@@@+@@@X@@@@kVAR@*@U8"SZ-7The total reactive power drawn form the source will be 7797@<7@:1@</7@<7@0@NormalTimes New Roman @@B@U`D@@@ p@@@ @@@@d@ Q.Source@@@@@@@d@Q.Load@@@@Q.Cap@@B@U@C@@@ p@@@@@@@d@ Q.Source@@@@@@@+@@@X@@@@kVAR@*@U`dmk- 79@<@:1@</@<@0@NormalTimes New Roman @*@U` */jl&&-@jAssuming the active power drawn by the load remains constant the apparent power drawn by the load will be 7j9j@@@ pA?@@A>A@@@{@A?AA@@A@ I.Load_oldAB@@A?AC@@+@AB@XAD@@ABAE@B@UX !$AF@@ pAG@@AFAH@@@AGAI@@dAHargAJ@@pAHAK@@AJ I.Load_oldAL@@AGAM@@+@AL@XAN@@ALdegAO*@U(1Qc5@))2)2-DNote: Comparing the magnitudes of the currents and power factors in both the cases (ie with and without capacitor bank connected in parallel with the load), we see that the load current is higher in the case without capacitor bank and this current is supplied by the source. Since the transmission lines connecting the source and load will have impedance, larger load currents will result in higher losses and larger voltage drops along the line and will ultimately result in poor voltage regulation. So if a capacitor is connected, in parallel with the load, it will meet some of reactive power required by the load and there by reduce the magnitude of the total current (or reactive power) that should be supplied by the source and hence the voltage drop in the line. Thus it provides better voltage regulation in heavily loaded lines. The same can be explained in terms of power factor. Loads operating at lower power factors will draw large reactive powers. So connecting a capacitor bank in parallel with the load will support some of the reactive power and there by improve the overall load ( overall load is parallel combination of load and capacitor) power factor and reduce the total load current.7AP<AQ8AOmArialAR<AS8AOmArialAPAT<AU8AOmArialARATAP9KKqAVA@@ pA@@ AA@@dAZ.bA@@AZ.aA@B@U0YtLh?A@@ pA@@ AA@@dAZ.cA@@5AA@@@AA@@dA1jA@@A0.9A@@AohmA*@UaGtp^"77- Part (a):7 A< A8AmTimes New Roman9 A< A:1A</ A< A0@NormalTimes New Roman A@B@UHf_A@@ pA@@ AA@@dAVaA@@5AA@@@AA@@tA1A@@AA@@dAeA@@5AA@@@AA@@dA1jA@@A0A@@AdegA@@AVA@B@UcbA@@ pA@@ AA@@dAVbA@@AA@@tA1A@@5AA@@@AA@@dAeA@@5AA@@@AA@@K@AA@@A1jA@@A120A@@AdegA@@AVA@B@UcA@@ pA@@ AA@@dAVcA@@AA@@tA1A@@5AA@@@AA@@dAeA@@5AA@@@AA@@dA1jA@@A120A@@AdegA@@AVA*@U`6nC&&-@ySince the system is not a balanced system we cannot apply per phase analysis. Applying KVL along different loops we have:7y9yA@@dB=I.aB?@@B=I.bB@@@B<I.cBA@@B;I.nBB*@U(Z,Y"22-we have79BC<BD:1BE</BF<BG0@NormalTimes New Roman BH@B@UhBI@@ pBJ@@,BIBK@@@BJBL@@@BKBM@@@BLBN@@dBMVaBO@@BMBP@@dBOI.aBQ@@pBOBR@@BQBS@@dBRjBT@@BR0.1BU@@BLBV@@dBUI.aBW@@BUZ.aBX@@BKBY@@p@BXBZ@@BYB[@@@BZB\@@dB[I.aB]@@B[I.bB^@@BZI.cB_@@pBXB`@@B_Ba@@dB`jBb@@B`0.1Bc@@BJ0Bd@B@UhBe@@ pBf@@,BeBg@@@BfBh@@@BgBi@@@BhBj@@dBiVbBk@@BiBl@@dBkI.bBm@@pBkBn@@BmBo@@dBnjBp@@Bn0.1Bq@@BhBr@@dBqI.bBs@@BqZ.bBt@@BgBu@@p@BtBv@@BuBw@@@BvBx@@dBwI.aBy@@BwI.bBz@@BvI.cB{@@pBtB|@@B{B}@@dB|jB~@@B|0.1B@@Bf0B@B@Uh4(B@@ pB@@,BB@@@BB@@@BB@@@BB@@dBVcB@@BB@@dBI.cB@@pBB@@BB@@dBjB@@B0.1B@@BB@@dBI.cB@@BZ.cB@@BB@@p@BB@@BB@@@BB@@dBI.aB@@BI.bB@@BI.cB@@pBB@@BB@@dBjB@@B0.1B@@B0B*@UPATjP -"Collecting terms we are left with:7"9"B<"B:1B</"B<"B0@NormalTimes New Roman B@B@UpixB@@ pB@@,BB@@@BB@@@BB@@@BB@@dBVaB@@BB@@dBI.aB@@pBB@@BB@@@BB@@dBjB@@B0.2B@@BZ.aB@@BB@@dBI.bB@@pBB@@BB@@dBjB@@B0.1B@@BB@@dBI.cB@@pBB@@BB@@dBjB@@B0.1B@@B0B@B@UpB@@ pB@@,BB@@@BB@@@BB@@@BB@@dBVbB@@BB@@dBI.aB@@pBB@@BB@@dBjB@@B0.1B@@BB@@dBI.bB@@pBB@@BB@@@BB@@dBjB@@B0.2B@@BZ.bB@@BB@@dBI.cB@@pBB@@BB@@dBjB@@B0.1B@@B0B@B@UpB@@ pB@@,BB@@@BB@@@BB@@@BB@@dBVcB@@BB@@dBI.aB@@pBB@@BB@@dBjB@@B0.1B@@BB@@dBI.bB@@pBB@@BB@@dBjB@@B0.1B@@BB@@dBI.cB@@pBB@@BB@@@BB@@dBjB@@B0.2B@@BZ.cB@@B0B*@U0 ? @ww->Three equations and three unknowns. Put this into matrix form:7>9>B<>B:1B</>B<>B0@NormalTimes New Roman B@B@U@$  X cB@@ pB@@,BB@@@BC@@p@BC@@0CC@@0ACC@@0ACC@@0ACC@@0ACC@@0ACC@@0ACC@@0ACC @@0ACC @@@C C @@C C @@5@C C @@@C C@@dC 1jC@@C 0.2C@@C ohmC@@C Z.cC@@5CC@@@CC@@dC1jC@@C0.1C@@CohmC@@5CC@@@CC@@dC1jC@@C0.1C@@CohmC@@5CC@@@CC@@dC1jC@@C0.1C @@CohmC!@@CC"@@5@C!C#@@@C"C$@@dC#1jC%@@C#0.2C&@@C"ohmC'@@C!Z.bC(@@5CC)@@@C(C*@@dC)1jC+@@C)0.1C,@@C(ohmC-@@5CC.@@@C-C/@@dC.1jC0@@C.0.1C1@@C-ohmC2@@5CC3@@@C2C4@@dC31jC5@@C30.1C6@@C2ohmC7@@CC8@@5@C7C9@@@C8C:@@dC91jC;@@C90.2C<@@C8ohmC=@@C7Z.aC>@@pBC?@@0C>C@@@0AC?CA@@0AC@CB@@@CACC@@CAI.cCD@@C@I.bCE@@C?I.aCF@@pBCG@@0CFCH@@0ACGCI@@0ACHCJ@@@CICK@@CIVcCL@@CHVbCM@@CGVaCN@B@U0  \ CO@@ pCP@@ COCQ@@p@CPCR@@0CQCS@@0ACRCT@@0ACSCU@@@CTCV@@CTI.cCW@@CSI.bCX@@CRI.aCY@@CPCZ@@@CYC[@@p@CZC\@@0C[C]@@0AC\C^@@0AC]C_@@0AC^C`@@0AC_Ca@@0AC`Cb@@0ACaCc@@0ACbCd@@0ACcCe@@@CdCf@@CdCg@@5@CfCh@@@CgCi@@dCh1jCj@@Ch0.2Ck@@CgohmCl@@CfZ.cCm@@5CcCn@@@CmCo@@dCn1jCp@@Cn0.1Cq@@CmohmCr@@5CbCs@@@CrCt@@dCs1jCu@@Cs0.1Cv@@CrohmCw@@5CaCx@@@CwCy@@dCx1jCz@@Cx0.1C{@@CwohmC|@@C`C}@@5@C|C~@@@C}C@@dC~1jC@@C~0.2C@@C}ohmC@@C|Z.bC@@5C_C@@@CC@@dC1jC@@C0.1C@@CohmC@@5C^C@@@CC@@dC1jC@@C0.1C@@CohmC@@5C]C@@@CC@@dC1jC@@C0.1C@@CohmC@@C\C@@5@CC@@@CC@@dC1jC@@C0.2C@@CohmC@@CZ.aC@@KCZC@@C1C@@pCYC@@0CC@@0ACC@@0ACC@@@CC@@CVcC@@CVbC@@CVaC@B@U(9 T LH C@@ pC@@CC@@{@CC@@CI.aC@@CC@@+@C@XC@@CC@B@U9 KT H eC@@ pC@@CC@@@CC@@dCargC@@pCC@@CI.aC@@CC@@+@C@XC@@CdegC@B@U4  ;h dC@@ pC@@CC@@p@CC@@0CC@@0ACC@@0ACC@@@CC@@CI.cC@@CI.bC@@CI.aC@@CC@@+@C@XC@@CC@B@U(Y t Mh C@@ pC@@CC@@{@CC@@CI.bC@@CC@@+@C@XC@@CC@B@UY Kt h fC@@ pC@@CC@@@CC@@dCargC@@pCC@@CI.bC@@CC@@+@C@XC@@CdegC@B@U(y  L C@@ pC@@CC@@{@CC@@CI.cC@@CC@@+@C@XC@@CC@B@Uy &  gC@@ pC@@CC@@@CC@@dCargC@@pCC@@CI.cC@@CC@@+@C@XC@@CdegC*@U0  A @-@HNotice that the currents are still fairly close to balanced three phase.7H9HC@@dD=1jD?@@D=1.0D@@@D<ohmDA@B@Ua | p +DB@@ pDC@@ DBDD@@dDCZ.c_2DE@@5DCDF@@@DEDG@@dDF1jDH@@DF1.0DI@@DEohmDJ*@UP - P #-#Applying per phase analysis we have7#9#DK<#DL:1DM</#DN<#DO0@NormalTimes New Roman DP@B@U8 _ jDQ@@ pDR@@ DQDS@@dDRI.a_2DT@@DRDU@@dDTVaDV@@DTDW@@dDVZ.a_2DX@@5DVDY@@@DXDZ@@dDY1jD[@@DY0.1D\@@DXohmD]@B@U s , 'D^@@ pD_@@D^D`@@{@D_Da@@D`I.a_2Db@@D_Dc@@+@Db@XDd@@DbDe@B@U   *Df@@ pDg@@DfDh@@@DgDi@@dDhargDj@@pDhDk@@DjI.a_2Dl@@DgDm@@+@Dl@XDn@@DldegDo@B@U8  ` kDp@@ pDq@@ DpDr@@dDqI.b_2Ds@@DqDt@@dDsI.a_2Du@@DsDv@@dDueDw@@5DuDx@@@DwDy@@K@DxDz@@Dy1jD{@@Dx120D|@@DwdegD}@B@U t - .D~@@ pD@@D~D@@{@DD@@DI.b_2D@@DD@@+@D@XD@@DD@B@U   /D@@ pD@@DD@@@DD@@dDargD@@pDD@@DI.b_2D@@DD@@+@D@XD@@DdegD@B@U8* L _@ lD@@ pD@@ DD@@dDI.c_2D@@DD@@dDI.a_2D@@DD@@dDeD@@5DD@@@DD@@dD1jD@@D120D@@DdegD@B@U1 sL ,@ 1D@@ pD@@DD@@{@DD@@DI.c_2D@@DD@@+@D@XD@@DD@B@U1 L @ 2D@@ pD@@DD@@@DD@@dDargD@@pDD@@DI.c_2D@@DD@@+@D@XD@@DdegD*@Ua Wt p `GG-7As a check, use the matrix equations set up for part A:7797D<7D:1D</7D<7D0@NormalTimes New Roman D@B@U(}  T D@@ pD@@ DD@@p@DD@@0DD@@0ADD@@0ADD@@@DD@@DI.cD@@DI.bD@@DI.aD@@DD@@@DD@@p@DD@@0DD@@0ADD@@0ADD@@0ADD@@0ADD@@0ADD@@0ADD@@0ADD@@0ADD@@@DD@@DD@@5@DD@@@DD@@dD1jD@@D0.2D@@DohmD@@DZ.c_2D@@5DD@@@DD@@dD1jD@@D0.1D@@DohmD@@5DD@@@DD@@dD1jD@@D0.1D@@DohmD@@5DD@@@DD@@dD1jD@@D0.1D@@DohmD@@DD@@5@DD@@@DD@@dD1jD@@D0.2D@@DohmD@@DZ.b_2D@@5DD@@@DD@@dD1jD@@D0.1D@@DohmD@@5DD@@@DD@@dD1jD@@D0.1D@@DohmD@@5DD@@@DD@@dD1jD@@D0.1D@@DohmD@@DD@@5@DD@@@DD@@dD1jD@@D0.2D@@DohmD@@DZ.a_2D@@KDE@@D1E@@pDE@@0EE@@0AEE@@0AEE@@@EE@@EVcE@@EVbE@@EVaE @B@U   D E @@ pE @@E E @@{@E E @@E I.aE@@E E@@+@E@XE@@EE@B@U   mE@@ pE@@EE@@@EE@@dEargE@@pEE@@EI.aE@@EE@@+@E@XE@@EdegE@B@U0 [ [0 E@@ pE@@EE@@p@EE@@0EE @@0AEE!@@0AE E"@@@E!E#@@E!I.cE$@@E I.bE%@@EI.aE&@@EE'@@+@E&@XE(@@E&E)@B@U ! < E0 E*@@ pE+@@E*E,@@{@E+E-@@E,I.bE.@@E+E/@@+@E.@XE0@@E.E1@B@U! < 0 nE2@@ pE3@@E2E4@@@E3E5@@dE4argE6@@pE4E7@@E6I.bE8@@E3E9@@+@E8@XE:@@E8degE;@B@U A \ DP E<@@ pE=@@E@@{@E=E?@@E>I.cE@@@E=EA@@+@E@@XEB@@E@EC@B@UA \ P oED@@ pEE@@EDEF@@@EEEG@@dEFargEH@@pEFEI@@EHI.cEJ@@EEEK@@+@EJ@XEL@@EJdeg