Circuits i i II EE221 Unit 11 Instructor: Kevin D. D Donohue Complex Power, Power Conservation, Power Factor Correction Correction, and Applications Complex Power Complex power has real and reactive components. Denote the sinusoidal voltage and current in a load by: Vˆ Vm v Iˆ I m i Then the complex power is expressed as: ˆIˆ* V V I Sˆ m v m i Vrms v I rms i Vrms I rms v i 2 2 2 Complex Power The real and imaginary terms of complex power represent the real (P) and reactive (Q) components of the power: Sˆ Vrms I rms v i Vrms I rms cos v i j Vrms I rms sin v i Sˆ P jQ Note that previously described power quantities can be obtained from complex power Apparent Power Sˆ S Vrms I rms units VAs Real (Average) Power ReSˆ P Vrms I rms cos v i units Watts Reactive Power ImSˆ Q Vrms I rms sin v i units VARs Power Factor P cos v i S Complex Power with Impedance Load impedance can be expressed as: ˆ V V Zˆ rms rms v i R jX Iˆrms I rms The above Th b relationship l ti hi can be b usedd to t express power in i terms of load impedance and either current or voltage magnitude. 2 * * Sˆ Vˆrms Iˆrms ZˆIˆrms I rms Zˆ Iˆrms * Vˆrms Vrms * ˆ ˆ ˆ ˆ S Vrms I rms Vrms * ˆ ˆ Z Z 2 Power Triangle The real and reactive terms of a load (R, (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 Impedance Power Triangle The power triangle provides a graphic representation of leading and lagging properties of the load: Q 0 Resistive Load Q 0 Capacitive Load (leading) Q 0 Inductive Load (lagging) Conservation of Power In 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: 0 N Sˆi i 1 Note that the above is only true for the real and reactive components. This is not true for apparent power. Power Factor Correction For a fixed generator voltage and load average power, 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). IˆS Ẑ Line VˆS Ẑˆ Load Ẑˆ C Power Factor Correction For 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 = x1 lagging and apparent power S1 = Irms Vrms that a new power factor of PF = x2 is achieved by placing a capacitor in parallel with the load (shunt) such that: S1 cos((1 )( )(tan((1 ) tan(( 2 )) C 2 Vrms 1 cos ( x2 ) for lagging 1 where 1 cos ( x1 ) and 2 1 cos ( x2 ) for leading Power Factor Correction For 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 = x1 leading and apparent power S1 = Irms Vrms that a new power factor of PF = x2 is achieved by placing a shunt inductor across the load such that: 2 Vrms L S1 cos(1 )(tan( 2 ) tan(1 )) 1 cos ( x2 ) for lagging 1 where 1 cos ( x1 ) and 2 1 cos ( x2 ) for leading Power Meters Power meters must simultaneously measure the voltage g ((in pparallel)) and the current (in series) associated with load of interest. interest The meter deflection is proportional i l the h average power. Electricity Consumption Cost The kilowatt-hours kilowatt hours (kWh) to a customer is measured with a kWh meter corresponding to the average power consumed over a period of time. (Energy Ch (E Charge)) Th The cost/rate t/ t off the th kWh may vary depending d di on when h 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 customer.