.MCAD 309000000 \  docDocument MmcObject[..  dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdjj shpRectV 3 mcDocumentObjectState\ mcPageModelK?>??mcHeaderFooterI@I c:\ee332\|F|N|D|P CHeaderFooterJ@Jc:\ee332\{f} {d}{n}@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 HeaderArial@PFooterArial@P Rotated Math FontTimes New Roman TextRegion* docRegionGshpBoxUH CharacterMap-RangeMap;An Introduction to Mathcad ChrPropMap7 RangeElem< ChrPropData8 RangeData=|0,0,128 ParPropMap9 <  ParPropData:@W?EmbedMap1 <LinkMap/ < LinkData0@NormalArial *@UB+(ZZ-W. H. Kersting7<8mArial9<:@W?1</<0@NormalArial *@U3fC@-New Mexico State University79<:@W?1</<0@NormalArial *@Ucmpe e -@Mathcad is a very valuable tool for all engineers. In many ways it is a "number crunching" word processor. This document will explain and demonstrate some of the basic features of Mathcad. 79<:@W?1</ <!0@NormalArial "*@U!  - Starting a new Mathcad document:7 #< $8"l Arial9 %< &:@W?1'</ (< )0@NormalArial **@U9k!!-AWhen you start a new document, do the following: 1. Click on File\New 2. Click on Blank Worksheet 3. Click on Format\Number A window is displayed. Set the following: "Displayed Precision" = 4 "Exponential Threshold" = 15 Click on j under "Imaginary" 4. Click OK 5. Click on File\Page Setup Set right margin = 0.5 in 6. Click OK79+<,:@W?1-</.</0@NormalArial 0*@Uq#-"Adding text to a Mathcad document:7"1<"280l Arial9"3<"4:@W?15</"6<"70@NormalArial 8*@U@@-A4Text can be added to a Mathcad document by first typing shift and " at the same time. This will present a small box on the screen. Text can now be typed into the box. When the text is completed hit Shift and then Enter. In all documents turned in, you must type in your name at the top of the document.74949<4::@W?1;</4<<4=0@NormalArial >*@UeUU- The Basics7 ?< @@8>l Arial9 @A< @B:@W?1@C</ @D< @E0@NormalArial @F*@UK+HC0C0-AAll of the usual math symbols are used in Mathcad, the only notation that might seem strange is the symbol used for the equal sign when a variable is defined. The symbol is the usual equal sign with a colon in front of the equal sign as demonstrated below:7@G<@H8@F0,0,128@I<@J8@FmArial0,0,128@G@KVP@@@ p@@@ @@@@d@C@@@@@@@d@A@@@@B@@B@UCVP@@@ p@@@@@@@d@C@@@@@@@+@@@X@@@@ _n_u_l_l_@*@U[{h|  -@A variable is raised to a power by typing the variable and then using the up arrow about the number 6. Hold shift and 6 at the same time.7@<@8@mArial9@<@:@W?1@</@<@0@NormalArial @*@Ud0LL- Square Root:7 @< @8@0,0,1289 @< @:@W?1@</ A< A0@NormalArial A@B@UA@@ pA@@ AA@@dACA@@{AA@@AAA@B@UA @@ pA @@A A @@dA CA @@A A @@+@A @XA@@A _n_u_l_l_A*@U-@EThe square root symbol above is achieved by pressing the slash key \.7EA@@tA=1jA?@@A=7A@@B@UH#p6V0AA@@ pAB@@AAAC@@dABEAD@@ABAE@@+@AD@XAF@@AD _n_u_l_l_AG*@UC}cPum q -@Note - the default for Mathcad is to use the operator i as the imaginary instead of j. To change the default to j, click on Format - Number and then click on the j.7AH<AI8AGmArial9AJ<AK:@W?1AL</AM<AN0@NormalArial AO*@Us="0% ) -@Note that when the value of D is displayed the mathematician's notation of putting the operator j after the imaginary part is used rather than in front as we do in electrical engineering.7AP<AQ8AO0,0,1289AR<AS:@W?1AT</AU<AV0@NormalArial AW*@Ui#0QQ-;All the typical mathematical operations can be carried out.7;AX<;AY8AW0,0,1289;AZ<;A[:@W?1A\</;A]<;A^0@NormalArial A_*@U([($033- Addition:7 A`< Aa8A_0,0,1289 Ab< Ac:@W?1Ad</ Ae< Af0@NormalArial Ag@B@U%Ah@@ pAi@@ AhAj@@dAiFAk@@AiAl@@dAkDAm@@AkEAn@B@U8&Ao@@ pAp@@AoAq@@dApFAr@@ApAs@@+@Ar@XAt@@Ar _n_u_l_l_Au*@U(n('0FF- Subtraction:7 Av< Aw8Au0,0,1289 Ax< Ay:@W?1Az</ A{< A|0@NormalArial A}@B@U(A~@@ pA@@ A~A@@dAFA@@AA@@dADA@@AEA@B@U8)A@@ pA@@AA@@dAFA@@AA@@+@A@XA@@A _n_u_l_l_A*@U(X(*000- Division:7 A< A8A0,0,1289 A< A:@W?1A</ A< A0@NormalArial A@B@U+A@@ pA@@ AA@@dAFA@@AA@@dADA@@AEA@B@U8,A@@ pA@@AA@@dAFA@@AA@@+@A@XA@@A _n_u_l_l_A*@U(x#( -0PP-Multiplication:7A<A8A0,0,1289A<A:@W?1A</A<A0@NormalArial A@B@U& .A@@ pA@@ AA@@dAFA@@AA@@dADA@@AEA@B@U8& /A@@ pA@@AA@@dAFA@@AA@@+@A@XA@@A _n_u_l_l_A*@U(3C(@00ll-Rasing to a power:7A<A8A0,0,1289A<A:@W?1A</A<A0@NormalArial A@B@U.F@1A@@ pA@@ AA@@dAFA@@AA@@dADA@@A2A@B@U38F@2A@@ pA@@AA@@dAFA@@AA@@+@A@XA@@A _n_u_l_l_A*@U(Stc(`30LL- Square Root:7 A< A8A0,0,1289 A< A:@W?1A</ A< A0@NormalArial A@B@UQf`4A@@ pA@@ AA@@dAFA@@AA@@{@AA@@ADA@@AEA@B@US8f`5A@@ pA@@AA@@dAFA@@AA@@+@A@XA@@A _n_u_l_l_A*@U{~P~P-ANote that in computing the square root above that when you type the \ key you get the square root symbol. Now type in the letter D. Note that the blue line stays just to the right of the letter D. If you continue to type the square root symbol will be carried over everything that you type. In the example above what you must do is after typing in the letter D, hit the space bar until the blue line is outside of the equare root symbol, now type the * and the letter E.79A<A:@W?1A</A<A0@NormalArial A*@UA60) - -@As demonstrated, all complex computations are carried out in rectangular form. To convert from rectangular to polar form the following can be done:7A<A8A0,0,1289A<A:@W?1A</A<A0@NormalArial A*@U0 07-Magnitude of a complex number:7A<A8A0,0,1289A<A:@W?1A</A<A0@NormalArial A@B@U  iG8A@@ pA@@ AA@@dADmagA@@{AA@@ADB@B@U 9B@@ pB@@BB@@dBDmagB@@BB@@+@B@XB@@B _n_u_l_l_B*@U#OC0}? ? -@The absolute magnitude brackets are achieved by pressing the | above the slash key (hold down the shift key and the slash \ at the same time).79B<B :@W?1B </B <B 0@NormalArial B *@U0Sc0`:-Angle of a complex number:7B<B8B 0,0,1289B<B:@W?1B</B<B0@NormalArial B@B@U JzqE`;B@@ pB@@ BB@@dBDangB@@BB@@@BB@@dBargB@@pBB@@BDB@@BdegB@B@USf`<B @@ pB!@@B B"@@dB!DangB#@@B!B$@@+@B#@XB%@@B# _n_u_l_l_B&*@U{(=0 -@Note that a Mathcad defined constant "deg" is used to determine the angle in degrees. The "arg" function computes the angle in radians. 7B'<B(8B&0,0,1289B)<B*:@W?1B+</B,<B-0@NormalArial B.@B@U@u[>B/@@ pB0@@B/B1@@dB0degB2@@B0B3@@+@B2@XB4@@B2 _n_u_l_l_B5*@U -@]All of the usual trig functions are used as follows. The angle must be specified in radians.7]9]B6<]B7:@W?1B8</]B9<]B:0@NormalArial B;@B@U@OB<@@ pB=@@ B@@dB=FB?@@B=B@@@{@B?BA@@B@EBB@@B?BC@@dBBsinBD@@pBBBE@@BDBF@@dBEDangBG@@BEdegBH@B@UBI@@ pBJ@@BIBK@@dBJFBL@@BJBM@@+@BL@XBN@@BLBO*@U` H B-@In the above after typing the E := hit shift and \ at the same time to get the brackets. Now type in the letter E. Once again you will need to hit the space bar to get the blue line outside of the brackets.79BP<BQ:@W?1BR</BS<BT0@NormalArial BU@B@U@.O(BV@@ pBW@@ BVBX@@dBWFBY@@BWBZ@@{@BYB[@@BZEB\@@BYB]@@dB\cosB^@@pB\B_@@B^B`@@dB_DangBa@@B_degBb@B@U.(Bc@@ pBd@@BcBe@@dBdFBf@@BdBg@@+@Bf@XBh@@BfBi@B@U@;NOHBj@@ pBk@@ BjBl@@dBkFBm@@BkBn@@{@BmBo@@BnEBp@@BmBq@@dBptanBr@@pBpBs@@BrBt@@dBsDangBu@@BsdegBv@B@U;NHBw@@ pBx@@BwBy@@dBxFBz@@BxB{@@+@Bz@XB|@@BzB}@B@U@[nVhB~@@ pB@@ B~B@@dBPFB@@BB@@dBcosB@@pBB@@BB@@dBargB@@pBB@@BDB@B@U[nhB@@ pB@@BB@@dBPFB@@BB@@+@B@XB@@BB*@U!?0  -@hTo define a complex number in polar form, the true mathematical expression for a complex number is used.7hB:@W?1C?</C@<CA0@NormalArial CB*@U m THmo-Matrix Algebra7CC<CD8CB| 0,0,1289CE<CF:@W?1CG</CH<CI0@NormalArial CJ*@U( 6 ( U  -@Matrices can be defined and operated upon in the usual manner. The notation that appears when a matrix is defined is shown below:79CK<CL:@W?1CM</CN<CO0@NormalArial CP@B@UX L d( CQ@@ pCR@@ CQCS@@dCRICT@@pCRCU@@0CTCV@@0ACUCW@@0ACVCX@@@CWCY@@CWCZ@@tCY15C[@@CYC\@@dC[eC]@@C[C^@@@C]C_@@tC^1jC`@@C^125Ca@@C]degCb@@CVCc@@tCb12Cd@@CbCe@@dCdeCf@@CdCg@@@CfCh@@K@CgCi@@Ch1jCj@@Cg115Ck@@CfdegCl@@CUCm@@tCl10Cn@@ClCo@@dCneCp@@CnCq@@tCp1jCr@@Cp0Cs*@U(c  (p eX@\@-A?Note - to create a matrix, select Insert-matrix and then define the dimension of the matrix. For some reason Mathcad defaults to starting matrice elements at a subscript of 0 rather than 1. This causes confusion for engineers. To change the default select Math/Options and then set "array origin" to 1 rather than 0.7^^?Ct<^Cu8CsCv<Cw8CsmArialCtCvCv9?Cx<D?:@W?1D@</DA<DB0@NormalArial DC@B@U( T : hDD@@ pDE@@DDDF@@@DEDG@@dDFVDH@@DFiDI@@DEDJ@@+@DI@XDK@@DIDL@B@UH q W iDM@@ pDN@@DMDO@@dDNVDP@@DNDQ@@+@DP@XDR@@DP _n_u_l_l_DS*@U0#300j-+Display the resulting matrix in polar form:7+9+DT<+DU:@W?1DV</+DW<+DX0@NormalArial DY@B@UH;zNWHkDZ@@ pD[@@ DZD\@@dD[nD]@@D[D^@@tD]1D_@@D]3D`@B@U8>a$PlDa@@ pDb@@DaDc@@@DbDd@@@DcDe@@dDdargDf@@pDdDg@@DfDh@@dDgVDi@@DgnDj@@DcdegDk@@DbDl@@+@Dk@XDm@@DkDn@B@UK`XmDo@@ pDp@@DoDq@@{@DpDr@@DqDs@@dDrVDt@@DrnDu@@DpDv@@+@Du@XDw@@DuDx*@U00n-To compute complex power:79Dy<Dz:@W?1D{</D|<D}0@NormalArial D~@B@U$oD@@ pD@@ DD@@@DD@@dDSD@@DnD@@DD@@@DD@@dDVD@@DnD@@DD@@@DD@@dDID@@DnD@B@U8fLpD@@ pD@@DD@@@DD@@dDSD@@DnD@@DD@@+@D@XD@@DD*@U00q-$The inverse of a matrix is found by:7$9$D<$D:@W?1D</$D<$D0@NormalArial D@B@UP.Fa@rD@@ pD@@ DD@@dDYD@@DD@@dDZD@@KDD@@D1D@B@U3F@sD@@ pD@@DD@@dDYD@@DD@@+@D@XD@@D _n_u_l_l_D*@U8kr{8xU::-@cNote - to raise a variable to a power, type the vaiable name Z and then the symbol above the 6 key.7c9cD  Root EntryHj1$@˜f@ Ole ContentsOlePres000   !"#$%&'()*+,-./01234"$  F  ^- "-   "- "-%"6"%/6%DF%FK%KO%OS%SW%W[%[] aija irsi rz{r z@= $ANGDIR0$ANGBASE0$AUNITS0$AUPREC2 $ORTHOSTEP90 $ORTHOBASE0$LUNITS2$LUPREC2 $LUNITSYS1$LUNAMEin $LUPAPER1in $LUPAPER2pt $LUPAPER3in $PSCALESYS Architectural$GDSPACE0 CONTINUOUS $INVISIBLE Y YSolid CONTINUOUS0 View_0 ,;AHA Previous??? - 0 BYLAYER   0  ף< CONTINUOUS-[$!@ʽ@ʃ!@ʽ@ h11`@t'@ #q!@t'@ C 0 Group_1  0 .}IG@jdo@It@0P z@ It@0P z@`K|@d71 @N!^h@R۸c> @N!^h@S۸c> @iAY@ >{ @iAY@ >{ @N!^h@۸c> @N!^h@$ 3+| @v#Q%@R+| @v#Q%@S+| @C(J@Y^ı @C(J@Y^ı @v#Q%@+| @v#Q%@ %  ף<Group_1 C0& Z  TCW40TEXTn@"M1T@Rn@"M1T@Sn@+y@n*,B@+y@n*,B@"M1T@n@"M1T@' 1 y{KD@Jˎ4@R y{KD@Jˎ4@S y{KD@[6@tʩ@[6@tʩ@Jˎ4@ y{KD@Jˎ4@C 0( Z"@cEs@R"@cEs@S"@0î)@t(\"@0î)@t(\"@cEs@"@cEs@) 4$d]"@eY,@R$d]"@eY,@S$d]"@2Ʃ&@}"@2Ʃ&@}"@eY,@$d]"@eY,@ *  ף<Group_4 C0+ E  TCW40TEXT:W7@&9l@R:W7@&9l@S:W7@bf@DRy@bf@DRy@&9l@:W7@&9l@, xPp"9@E38@RPp"9@E38@SPp"9@^]@S@^]@S@E38@Pp"9@E38@C 0- -|@Yoiq@R|@Yoiq@S|@%<޸@'F@%<޸@'F@Yoiq@|@Yoiq@. +7w)@C˅b@R7w)@C˅b@S7w)@b͇@+0@b͇@+0@C˅b@7w)@C˅b@ /  ף<Group_1 C00 ?@Arm@Ws@^B@I@ճ @ZD@zݠ@  01 ?@?dH@>@ \@2 ?@j<@UsN8@[3@ C 03 Group_1 C04 Z @}@Fbu@enf@9@G0.@E @Xl@  05 Z @$l@<] @MPl@6 Z @p= Oa@¢ @kC@ C 07 Group_3 C08 I  TCW40TEXT6$<@K@R6$<@K@S6$<@xc@Fp'@xc@Fp'@K@6$<@K@9 183@Ȁ@R83@Ȁ@S83@{@K<@{@K<@Ȁ@83@Ȁ@C 0: IHw@@RHw@@SHw@g1@c?!@g1@c?!@@Hw@@; 2Q-@i@RQ-@i@SQ-@i@#@i@#@i@Q-@i@ <  ף<z%]a%@D%%6D;F%;F2K%2K;O%;O2S%2S;W%;W2[%2[6]%6]6a%6@6D%66%R"T%TX&%X&\%\a&%a&e%ei&%i&k" "o"w'xo "w"'w ""' ""'%k"o"%N"R"%""%6l6%6`6d%.d?d l.l?Ch*%6i6l%"%&%&%&%&%&%&" ""' ""' "" '  " "' %""%""%""%/"P"%""%6"6C%66%"C%eAC%/"/C%/d/ "- Times New Roman-!ZSCourier New- Times New Roman-!2Y- Times New Roman-!Z- Times New Roman-!3- "- "- Times New Roman-!Zg- Times New Roman-!1 r- Times New Roman-!ZQE- Times New Roman-!4WQ- "- "- Times New Roman-!EE- Times New Roman-!xL- Times New Roman-!-^ - Times New Roman-!+1- "- 9h{i|9I%jzrs%j{r 6wx6%wp%x~ "- Times New Roman-!IlR- Times New Roman-!1rY- Times New Roman-!Ij- Times New Roman-!2p---D*@U V+ 6 6 -@For this example, we want to compute the two loop currents I1 and I2 given the source voltage and all of the impedances. "Notational" subscripts will be used to set up the matrix equation.79D<D:@W?1D</D<D0@NormalArial D*@U CDS P$$-Given:79D<D:@W?1D</D<D0@NormalArial D@B@U`>YtPD@@ pD@@ DD@@dDE.xD@@DD@@tD100D@@DD@@dDeD@@DD@@tD1jD@@D0D@B@UC YPD@@ pD@@ DD@@dDZ.1D@@DD@@tD2D@@DD@@tD1jD@@D3D@B@U(CtY=PD@@ pD@@ DD@@dDZ.2E@@DE@@tE5E@@EE@@tE1jE@@E8E@B@UCYPE@@ pE@@ EE@@dEZ.3E @@EE @@tE 1E @@E E @@tE 1jE @@E 2E@B@UCSYPE@@ pE@@ EE@@dEZ.4E@@EE@@tE3E@@EE@@tE1jE@@E5E*@U sS 30;0-@Solution: Step 1: Define the source matrix by inserting a 2x1 matrix. Enter the source voltages in the two loops. No source in loop 2.79E<E:@W?1E</E<E0@NormalArial E@B@UXE@@ pE@@ EE @@dE E.sourceE!@@pEE"@@0E!E#@@0AE"E$@@@E#E%@@E#0E&@@E"E.xE'*@U8 8E-@EStep 2: Define the 2x2 loop impedance matrix and enter the elements.7E9EE(@@KE3E?@@E>Z.2E@@@E2EA@@dE@Z.1EB@@E@Z.2EC*@U8[k8hm-@UStep 3: Solve the matrix equation by multiplying the Zloop inverse times the source.7U9UED  Root EntryHj1$@t?f@Ole ContentsG!OlePres000  !"#$%&'()*+,-./0123456789:;<=>?@"$   ^- "-   "- Times New Roman-!+0Courier New- Times New Roman-!-^ - "- @V $ANGDIR0$ANGBASE0$AUNITS0$AUPREC2 $ORTHOSTEP90 $ORTHOBASE0$LUNITS2$LUPREC2 $LUNITSYS1$LUNAMEin $LUPAPER1in $LUPAPER2pt $LUPAPER3in $PSCALESYS Architectural$GDSPACE0 CONTINUOUS $INVISIBLE Y YSolid CONTINUOUS0 View_0 ,;AHA Previous??? 5 0BYLAYER C 0 CONTINUOUS Times New Roman+  TCW40TEXTnD@Ce #@RnD@Ce #@SnD@bH@Q'S@bH@Q'S@Ce #@nD@Ce #@-@@YoIQ1@R@@YoIQ1@S@@Jx1@O^m-x@Jx1@O^m-x@YoIQ1@@@YoIQ1@  ף<Group_4 C0E  TCW40TEXTux@&@@Rux@&@@Sux@bie@쥶tj@bie@쥶tj@&@@ux@&@@xD@E@RD@E@SD@^c@ |Ha @^c@ |Ha @E@D@E@ C 0 ף<Group_2  0SSE @,!۳@/U@,!۳@h@I_TV@/U@,!۳@/U@,!۳@h@@ Q @,!۳@SSE @,!۳@ C 0Group_3 C0I  TCW40TEXTӕB@hleآ@RӕB@hleآ@SӕB@59@r@59@r@hleآ@ӕB@hleآ@1)?"@G_@R)?"@G_@S)?"@Ti@"$C@Ti@"$C@G_@)?"@G_@ C 0 ף<Group_2  0Uzgj@E-@(|@E-@"G47@x`7!V@(|@E-@(|@E-@"G47@к@[@E-@Uzgj@E-@ C 0 3  TCW40TEXT4 }U@zM!R@R4 }U@zM!R@S4 }U@GQw@|D@GQw@|D@zM!R@4 }U@zM!R@IH_@ed@RH_@ed@SH_@2O^@5N{I@2O^@5N{I@ed@H_@ed@4"ų@u- @R"ų@u- @S"ų@\ @Wp2@\ @Wp2@u- @"ų@u- @Iv5e@ČH@Rv5e@ČH@Sv5e@XV)d @Lod@XV)d @Lod@ČH@v5e@ČH@2B ė@rZ@RB ė@rZ@SB ė@ \\ @[/{@ \\ @[/{@rZ@B ė@rZ@IUa?F@nSC@RUa?F@nSC@SUa?F@27 @7y`w@27 @7y`w@nSC@Ua?F@nSC@   ף<Group_2  0D I@hyN@D I@j`6`@xC?|@7-Zm@D I@j`6`@D I@j`6`@ݛ|@7-Zm@D I@D}@D I@hyN@ C 0Group_2  0|s@ݛN@|s@Tƣ6`@@ pm@|s@Tƣ6`@ |s@Tƣ6`@I@~@ pm@ |s@w5@|s@ݛN@ C 0 4  TCW40TEXTI@e9e@RI@e9e@SI@ReN@.BM@ReN@.BM@e9e@I@e9e@ ZeBp@cES@ReBp@cES@SeBp@a$F@@a$F@@cES@eBp@cES@   ף<Group_1 C0 Z  TCW40TEXTe6Q@"-j@Re6Q@"-j@Se6Q@}d9@n*s@}d9@n*s@"-j@e6Q@"-j@ 1 y{u@Jˎ@R y{u@Jˎ@S y{u@[d@th@[d@th@Jˎ@ y{u@Jˎ@C 0 3| )@v#Q@R| )@v#Q@S| )@C @I@C @I@v#Q@| )@v#Q@ Zq7m@N!ї(@Rq7m@N!ї(@Sq7m@i!M@r-3<'@i!M@r-3<'@N!ї(@q7m@N!ї(@ 26Ɔm@7n@R6Ɔm@7n@S6Ɔm@hmc&@كw@hmc&@كw@7n@6Ɔm@7n@ Z+@5@R+@5@S+@s@~gk@s@~gk@5@+@5@   0  ף<fff'( @L@fff'( @ffO @ fff'( @@fff'( @33s'b@ C 0 fff'( @?Y@h @?Y@h @?Y@fff'( @33s'b@   0 yz@L* @yz@ffO @ yz@@yz@33s'b@ fffF@L* @fffF@ffO @ fffF@@fffF@33s'b@ fffF@@233@@ fff'( @@233 @@ C 0 Group_1  0 \@\:N@GT@(q@ GT@(q@P!@5mׁ@ P!@5mׁ@\#@(q@ \#@(q@)P@5mׁ@ )P@5mׁ@Qz|@(q@ Qz|@(q@GTé@5mׁ@ GTé@5mׁ@)@\:N@ C0 GT@\:N@P!F@\:N@Qz@\:N@GT@(q@ \s@\:N@)@\:N@P!F@\:N@\s@(q@ t2@Ji@~A@Ji@6e@Ji@t2@6q@ NR&@Ji@{XS@Ji@L@Ji@NR&@6q@  0 )@\:N@Qz@\:N@! P!@\:N@\@\:N@" \S@\:N@)@\:N@ 0# :t@L~@:t@AB@ C 0$ :t@n6HΉ@*ӈM @Fl|J@obl@1*G@<; g@)h;@   0% uPj@ꠎP@333U "@ꠎP@& BG@$"@BG@X|E*@' fffF@ffOD@fffF@L* @( fffF@L@fffF@L@ C 0) Group_1  0* gh$@RL8@4a5Q@p@+ 4a5Q@p@C"@ׅk@, C"@ׅk@gh@p@- gh@p@v.V^@ׅk@. v.V^@ׅk@Ǜ@p@/ Ǜ@p@U@ׅk@0 U@ׅk@+@RL8@ C01 Ux@RL8@"g}g@RL8@K@RL8@Ux@p@2 3J4^@RL8@+@RL8@"g}g@RL8@3J4^@p@3 93-@A{@ڇZ@A{@l8Af@A{@93-@^H@4 l@A{@R9LY@A{@'X@A{@l@^H@  05 +@RL8@K@RL8@6 C" @RL8@gh$@RL8@7 3J4^@RL8@+@RL8@ 08 q5I@BMQa @q5I@LJ @9 q5I@?su}@q5I@r!P@: q5I@T%C@q5I@هX @; h|@ @BZ@q5I@T%C@< >@هX @h|@ @BZ@= h|@T%۳@>@هX @> >@r!@h|@T%۳@? h|@?su @>@r!@@ >@ @B:@h|@?su @A q5I@r!P@>@ @B:@ C 0B Group_1  0C .M x@jdP@פ@00B:@D פ@00B:@`K=K@d71du @E `K=K@d71du @פ@jd@F פ@jd@`K=K@ʝ۳@G `K=K@ʝ۳@פ@@H פ@@`K=K@00BZ@I `K=K@00BZ@.M x@ʝC@ C0J .M x@`a@.M x@nbz@.M x@@פ@`a@K .M x@..Q!@.M x@;/@.M x@nbz@פ@..Q!@L GMw@2A!k@GMw@ @GMw@2ں?@vl@2A!k@M GMw@d^ @GMw@fZa @GMw@G@vl@d^ @  0N .M x@ʝC@.M x@@O .M x@d71du}@.M x@jdP@P .M x@..Qa @.M x@;/ @ 0Q b:! @, @G|@, @R ZLg-xy@@5@@ C 0S Solidb:{}@YL@^@YL@^@YL@b:{}@@T @P@>@P@>@P@@>B@ U  ף<Solid"- Times New Roman-!EE- Times New Roman-!xL- "-%O2t2%h,t2%t2h9%=2O2 "- Times New Roman-!I%{- Times New Roman-!1+- "-%22%,2%29%22 "- Times New Roman-!3,- Times New Roman-!I& - Times New Roman-!4}- Times New Roman-!Iw- Times New Roman-!2y- Times New Roman-!Is- "-%Qv%$jv%vj%?Q%Qv%jv%vj%?Q "- Times New Roman-!4WP- Times New Roman-!ZQE- "- "- Times New Roman-!Zg- Times New Roman-!1 r- Times New Roman-!3- Times New Roman-!Z- Times New Roman-!2Y- Times New Roman-!ZR- "-%/d/%/"/CeAC%%"C%66%6"6C%""%/"P"%"%&%&%&%&%&%&" ""' ""' "" '  " "' %""%""%""%6i6l l.k>Ch*%.d?d%6`6d%6l6%R"T%TX&%X&]%]a&%a&e%ei&%i&k" "o"w'xo "w"'w ""' ""'%k"o"%N"R"%""%66%6@6D%6]6a%2[6]%:W2[%2S:W%:O2S%2J:O%:F2J%6D:F%DF%FJ%JO%OS%SW%W[%[] aija iqri rz{r zz%]a%@D%%/6%"5"- "-'--'- "-&:2--&:2--E*@U+( -@LThis time the impedances will be definned using variable subscript notation.7L9LE0@NormalArial F?@B@UP1`F@@@ pFA@@ F@FB@@dFACFC@@pFAFD@@0FCFE@@0AFDFF@@0AFEFG@@0AFFFH@@0AFGFI@@0AFHFJ@@0AFIFK@@0AFJFL@@0AFKFM@@0AFLFN@@0AFMFO@@0AFNFP@@0AFOFQ@@0AFPFR@@0AFQFS@@0AFRFT@@@FSFU@@FS1FV@@FR0FW@@FQFX@@dFWzFY@@FW4FZ@@FP0F[@@KFOF\@@F[1F]@@FN1F^@@FMF_@@dF^zF`@@F^3Fa@@FL0Fb@@FK0Fc@@FJ1Fd@@KFIFe@@FdFf@@dFezFg@@Fe2Fh@@FHFi@@dFhzFj@@Fh2Fk@@FG0Fl@@KFFFm@@Fl1Fn@@FE0Fo@@FDFp@@dFozFq@@Fo1Fr@B@U!Fs@@ pFt@@FsFu@@dFtCFv@@FtFw@@+@Fv@XFx@@FvFy@B@UY1-#Fz@@ pF{@@ FzF|@@dF{SourceF}@@pF{F~@@0F}F@@0AF~F@@0AFF@@0AFF@@@FF@@F0F@@F0F@@F0F@@F~E.xF*@U([k(h}-Solve for the branch currents:79F<F:@W?1F</F<F0@NormalArial F@B@UPvF@@ pF@@ FF@@dF I.branchF@@FF@@@FF@@dFCF@@KFF@@F1F@@FSourceF*@U((}-&Display branch currents in polar form:7&9&F<&F:@W?1F</&F<&F0@NormalArial F@B@U R/$F@@ pF@@ FF@@dFnF@@FF@@tF1F@@F4F@B@UP%F@@ pF@@FF@@{@FF@@FF@@dF I.branchF@@FnF@@FF@@+@F@XF@@FF@B@U&F@@ pF@@FF@@@FF@@@FF@@dFargF@@pFF@@FF@@dF I.branchF@@FnF@@FdegF@@FF@@+@F@XF@@FF*@U(O('}'( -/Compute the branch voltages and complex powers:7/9/F