ECE 423Power Systems AnalysisThree-PhaseReviewS01Erms13.856 kV⋅:= γ 0 deg⋅:= j1−:=Instantaneousf50Hz⋅:=ω 2 π⋅ f⋅:= ω 314.159rads⋅=T1f:=T 0.02 s= Tθγω:=eant() 2Erms⋅ cos ω t⋅ γ+()⋅:=0 0.01 0.02 0.03 0.042− 104×1− 104×01104×2104×"A" Phase Internal VoltageTime (sec)Voltage (V)eant()tC:\JoeLaw\Classes\ECE 423 Spring 2009\Handouts\General\s01b Three Phase.xmcdPage 1 of 9 January 14, 2009ECE 423Power Systems AnalysisThree-PhaseReviewS01Phasor RepresentationEanErmsej γ⋅⋅:=030609012015018021024027030033005103×1104×1.5 104×Phasor Diagram of the "A" Phase VoltageEanarg Ean()C:\JoeLaw\Classes\ECE 423 Spring 2009\Handouts\General\s01b Three Phase.xmcdPage 2 of 9 January 14, 2009ECE 423Power Systems AnalysisThree-PhaseReviewS01Three-Phaseσ2 π⋅3:= σ 2.094 rad⋅= σ 120 deg⋅=ebnt() 2Erms⋅ cos ω t⋅ γ+ σ−()⋅:= ecnt() 2Erms⋅ cos ω t⋅ γ+ σ+()⋅:=EbnErmsej γσ−()⋅⋅:=EcnErmsej γσ+()⋅⋅:=0 0.0033 0.0067 0.01 0.0133 0.0167 0.022− 104×1− 104×01104×2104×Internal Phase VoltagesTime (sec)Voltage (V)eant()ebnt()ecnt()tC:\JoeLaw\Classes\ECE 423 Spring 2009\Handouts\General\s01b Three Phase.xmcdPage 3 of 9 January 14, 2009ECE 423Power Systems AnalysisThree-PhaseReviewS01030609012015018021024027030033005103×1104×1.5 104×Phasor Diagram of the VoltagesEanEbnEcnarg Ean()arg Ebn(), arg Ecn(), C:\JoeLaw\Classes\ECE 423 Spring 2009\Handouts\General\s01b Three Phase.xmcdPage 4 of 9 January 14, 2009ECE 423Power Systems AnalysisThree-PhaseReviewS01r10Ω⋅:=L 10 mH⋅:= XLω L⋅:= XL3.142 Ω=ZrjXL⋅+:= Z 10 3.142i+()Ω=Z 10.482 Ω= arg Z( ) 17.441 deg⋅=θ arg Z():= θ 0.304= θ 17.441 deg⋅=IaEanZ:= Ia1.261 0.396i−()kA⋅= Ia1.322 kA⋅= arg Ia()17.441− deg⋅=IbEbnZ:= Ib0.974− 0.894i−()kA⋅= Ib1.322 kA⋅= arg Ib()137.441− deg⋅=IcEcnZ:= Ic0.287− 1.29i+()kA⋅= Ic1.322 kA⋅= arg Ic()102.559 deg⋅=InIaIb+ Ic+:= In0kA⋅=C:\JoeLaw\Classes\ECE 423 Spring 2009\Handouts\General\s01b Three Phase.xmcdPage 5 of 9 January 14, 2009ECE 423Power Systems AnalysisThree-PhaseReviewS01030609012015018021024027030033005103×1104×1.5 104×Phasor Diagram of the Voltages and CurrentsEanEbnEcnIa10⋅Ib10⋅Ic10⋅arg Ean()arg Ebn(), arg Ecn(), arg Ia(), arg Ib(), arg Ic(), C:\JoeLaw\Classes\ECE 423 Spring 2009\Handouts\General\s01b Three Phase.xmcdPage 6 of 9 January 14, 2009ECE 423Power Systems AnalysisThree-PhaseReviewS01iat()2Erms⋅Zcos ω t⋅ γ+ θ−()⋅:= ibt()2Erms⋅Zcos ω t⋅ γ+ θ− σ−()⋅:= ict()2Erms⋅Zcos ω t⋅ γ+ θ− σ+()⋅:=0 0.0033 0.0067 0.01 0.0133 0.0167 0.022− 104×1− 104×01104×2104×2000−1000−010002000Voltages and CurrentsTime (sec)Voltage (V)eant()ebnt()ecnt()iat()ibt()ict()tC:\JoeLaw\Classes\ECE 423 Spring 2009\Handouts\General\s01b Three Phase.xmcdPage 7 of 9 January 14, 2009ECE 423Power Systems AnalysisThree-PhaseReviewS01030609012015018021024027030033005103×1104×1.5 104×Phasor Diagram of the Voltages and CurrentsEanIa10⋅arg Ean()arg Ia(), C:\JoeLaw\Classes\ECE 423 Spring 2009\Handouts\General\s01b Three Phase.xmcdPage 8 of 9 January 14, 2009ECE 423Power Systems AnalysisThree-PhaseReviewS010 0.0033 0.0067 0.01 0.0133 0.0167 0.022− 104×1− 104×01104×2104×2000−1000−010002000Voltages and CurrentsTime (sec)Voltage (V)eant() iat()tC:\JoeLaw\Classes\ECE 423 Spring 2009\Handouts\General\s01b Three Phase.xmcdPage 9 of 9 January 14,