1EE2315Lecture9Phasor CircuitAnalysis,EffectiveValueandComplexPower:Watts,VAR’sandVolt‐AmperesEffectiveValueofaSinusoid(1/2)Average Power:i(t) 10 2In our example:Also:The effective value is also called the Root Mean Square value or rms value.EffectiveValueofaSinusoid(2/2)R‐CCircuitExample(1/6)Capacitive Reactancevs(t)40 88.42 F+ vR -+vc-i3Using rms phasor for voltage source.R‐CCircuitExample(2/6)12040 -j30 + VR -~+VC-~I~Calculate Real Power:And Reactive Power:Apparent power is the product of voltage and current of the source.Also:R‐CCircuitExample(3/6)4Power Factor is the ratio of real power to apparent power:Power Factor is also the Cosine of the anglebetween the load voltage and the load current:If the load current leads the load voltage, the power factor is leading; if it lags the load voltage,the power factor is lagging.R‐CCircuitExample(4/6)Phasor Diagram of Voltage and CurrentCurrent leads voltage.R‐CCircuitExample(5/6)36.87oI~~V120V2.4A5The Power Triangle showing leadingpower factor.R‐CCircuitExample(6/6)CalculatingComplexPower(1/2)2SPjQIRjX  *2*22,I c jd I c jdthen I I c d I  *SPjQIIRjX  RjXI~+ V -~cjdI~6CalculatingComplexPower(2/2)VRjXI*SPjQVI From now on, we use the above method to calculate complex power.LaggingPowerFactorExample(1/4)v(t)5 22.97 mHi7Calculate complex power directly:LaggingPowerFactorExample(2/4)1200°5 j8.66 I~Power Factor:Power Factor is LaggingPhasor Diagram of Voltage and CurrentLaggingPowerFactorExample(3/4)60o1200° V12-60° Aimagreal8Power Triangle for lagging Power FactorLaggingPowerFactorExample(4/4)1440 VA720 W1247 VAR'simagrealPhasorPowerExample(1/4)480V(rms)0.5 j2 40 j30 -j150 +V-~I~480 07.94 20.08(56.75 20.75)VIAj(40 30)( 150)(56.25 18.75)40 30 150pjjZjjj9PhasorPowerExample(2/4)3.550.949 94.9%3.74PF or Lagging*(471 1.65 )(7.94 20.08 )3.55 1.18RLCSVIVAkW j kVAR (56.25 18.75) 471 1.65VI j    PhasorPowerExample(3/4)*(471 1.65 )(9.42 38.52 )3.55 2.66RL RLSVIVAkW j kVAR 471 1.659.42 38.52(40 30)RLVIAjCapacitorVA R ’s:224711.48150cVQ