iiCircuits IIEE221EE221Unit 11Instructor: Kevin D DonohueInstructor: Kevin D. DonohueComplex Power, Power Conservation, Power Factor Correction and ApplicationsFactor Correction, and ApplicationsComplex PowerComplex Power Complex power has real and reactive components. Denote the sinusoidal voltage and current in a load by:imvmIIVVˆ ˆ Then the complex power is expressed as:imvmVVˆˆ* ivrmsrmsirmsvrmsimvmIVIVIVIVS222ˆ*Complex PowerComplex Power The real and imaginary terms of complex power represent the real (P) and reactive (Q) components ofrepresent the real (P) and reactive (Q) components of the power: IVjIVIVSiiisincosˆNote that previously described power quantities can bejQPSIVjIVIVSivrmsrmsivrmsrmsivrmsrmsˆsincosNote that previously described power quantities can be obtained from complex power rmsrmsIVSS VAs units ˆPowerApparent ivrmsrmsivrmsrmsPIVQSIVPSVARs units sinˆImPower Reactive Watts units cosˆRePower (Average) RealivSPcosFactorPowerComplex Power with ImpedanceComplex Power with Impedance Load impedance can be expressed as:Th b l ti hi b d t ijXRIVIVZivrmsrmsrmsrmsˆˆˆThe above relationship can be used to express power in terms of load impedance and either current or voltage magnitude. 2ˆˆˆˆˆˆˆ2*2**ˆˆˆˆˆˆˆˆVIZIIZIVSrmsrmsrmsrmsrms**ˆˆˆˆˆˆZVZVVIVSrmsrmsrmsrmsrmsPower TrianglePower TriangleThe real and reactive terms of a load (RX) can be represented by aThe real and reactive terms of a load (R, X) can be represented by a triangle modeling the vector addition. The legs of the triangle are the real (P) and reactive (Q) components and complex power:Phase of ImpedancePower TrianglePower TriangleThe power triangle provides aThe power triangle provides a graphic representation of leading and lagging properties of the load:of the load:(leading)LoadCapacitive0QLoad Resistive0Q(lagging) Load Inductive0Q(leading)LoadCapacitive0QConservation of PowerConservation of PowerIn a given circuit the complex power absorb (denoted byIn a given circuit the complex power absorb (denoted by positive values) equals the complex power delivered (denoted by negative values).For a circuit with N elements the sum of all power is zero:NNiiS1ˆ0Note that the above is only true for the real and reactive components. This is not true for apparent power.Power Factor CorrectionPower Factor CorrectionFor a fixed generator voltage and load average power the outputFor a fixed generator voltage and load average power, the output current should be minimized to limit losses over the power line. This is done by adding reactive components to the power systems to bring the PF to 1 (or close to it).bring the PF to 1 (or close to it).SIˆSLineZˆˆˆSVˆLoadZˆCZˆPower Factor CorrectionPower Factor CorrectionFor an inductive load (PF lagging) a purely capacitive load can beFor an inductive load (PF lagging) a purely capacitive load can be added to the line to bring the power factor closer to 1. Show that for a load with PF = x1lagging and apparent power S1= IrmsVrmsthat a new power factor of PF=x2is achieved by placing a capacitor in parallelpower factor of PF x2is achieved by placing a capacitor in parallel with the load (shunt) such that:))tan())(tan(cos(2111SClaggingfor)(cos))())( ((1122111xVCrmsleadingfor )(coslaggingfor )(cos and )(cos where2122111xxxPower Factor CorrectionPower Factor CorrectionFor a capacitive load (PF leading) a purely inductive load can be addedFor a capacitive load (PF leading) a purely inductive load can be added to the line to bring the power factor closer to 1. Show that for a load with PF = x1leading and apparent power S1= IrmsVrmsthat a new power factor of PF=x2is achieved by placing a shunt inductor acrosspower factor of PF x2is achieved by placing a shunt inductor across the load such that:2Vlaggingfor)(cos))tan())(tan(cos(11211xSVLrmsleadingfor )(coslaggingfor )(cos and )(cos where2122111xxxPower MetersPower MetersPower meters must simultaneously measure the voltage (in parallel) g( p )and the current (in series) associated with load ofinterestinterest. The meter deflection is ilhproportional the average power.Electricity Consumption CostElectricity Consumption CostThe kilowatthours (kWh) to a customer is measured with a kWh meterThe kilowatt-hours (kWh) to a customer is measured with a kWh meter corresponding to the average power consumed over a period of time.(E Ch ) Th t/ t f th kWh d di h(Energy Charge) The cost/rate of the kWh may vary depending on when the power is used (high vs. low demand), and how much total power has been consumed (cost may go down after so many kWh used).(Demand Charge) A fixed overhead amount is charged simply to maintain the power delivery system, even if you use no power.A penalty may also be imposed for having a pf below a set figure (i.e. 0.9) since it requires larger currents and the unmetered losses in the line to the
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