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Power in AC Circuits ELEC 308 Elements of Electrical Engineering Dr. Ron Hayne Images Courtesy of Allan Hambley and Prentice-Hall Power Delivered to a Load V Vm0o Vm I Im where Im Z Z Z ELEC 308 2 RESISTIVE Load Pure Resistance Current in Phase with Voltage Z R R0 vt Vm cost i t I m cost pt vt i t Vm I m cos 2 t ELEC 308 3 INDUCTIVE Load Pure Inductance Current Lags Voltage Reactive power flows from source to load; Pavg = ______ Z L jL L90 vt Vm cost i t I m cos t 90 I m sin t pt vt i t Vm I m cost sin t ELEC 308 4 CAPACITIVE Load Pure Capacitance Current Leads Voltage Reactive power flows from source to load; Pavg = ______ 1 1 ZC 90 jC C vt Vm cost i t I m cos t 90 I m sin t pt vt i t Vm I m cost sin t ELEC 308 5 Importance of Reactive Power No average power is consumed by a pure energy-storage element Reactive power still important Transmission lines, transformers, fuses, etc. must be able to withstand the current associated with reactive power Possible to have loads that draw LARGE currents, even though little average power is consumed Electric power companies STILL charge for reactive power (at a lower rate), as well as total energy delivered ELEC 308 6 Real Power Z RLC Z RLC vt Vm cost it I m cost Vm I m V I cos m m cos 2 2 2 Vrms I rms cos P Units of average (REAL) power P are in watts (W). ELEC 308 7 Power Factor The term cos(θ) is called the power factor: PF = cos(θ) The power angle θ is taken as the phase of the voltage θv minus the phase of the current θi θ = θv-θi Current lags voltage = POSITIVE power angle Current leads voltage = NEGATIVE power angle Sometimes stated as a PERCENTAGE e.g. 90% lagging => cos(θ) = 0.9 and Curr. lags Volt. ELEC 308 8 Reactive Power Capacitance voltage increasing/decreasing Energy flowing into/out of capacitance Inductance current increasing/decreasing Energy flowing into/out of inductance Instantaneous power can be VERY large Average power (and net energy) is still _________ Reactive power is peak instantaneous power associated with energy-storage elements Q = VrmsIrmssin(θ) Units are VARs (Volt Amperes Reactive) ELEC 308 9 Apparent Power Apparent power is the product of the effective voltage and effect current S = VrmsIrms Units are volt-amperes (VA) Can be determined from real and reactive powers: S P Q 2 ELEC 308 2 10 Units Units indicate whether quantity is power, reactive power, or apparent power 5-kW load means that P = 5 kW 5-kVA load means that VrmsIrms = 5 kVA 5-kVAR load means that Q = 5 kVAR ELEC 308 11 Power Triangle Demonstrates relationships between Real power, P Reactive power, Q Apparent power VrmsIrms Power angle, θ ELEC 308 12 Additional Power Relationships PI 2 rms R 2 rms V P R 2 Q I rms X 2 Vrms Q X ELEC 308 13 Using Power Triangles Find the power, reactive power, and power factor for the source in the circuit below. ELEC 308 14 Using Power Triangles ELEC 308 15 Using Power Triangles ELEC 308 16 Power Factor Correction In heavy industry Many loads are partly inductive = large amounts of reactive power flow Causes higher current in transmission system Energy rates charged to industry depend on the power factor Higher charges for energy delivered at lower power factors Advantageous to choose loads that operate at near unity power factor Common approach is to place capacitors in parallel with an inductive load to increase the power factor ELEC 308 17 Thévenin and Norton Use same techniques for circuits with impedances as we did for resistive circuits ELEC 308 18 Maximum Power Transfer ELEC 308 19 Maximum Power Transfer Maximum power transferred achieved by maximizing the current First case: Load is complex impedance Load impedance for max. power transfer is Z load Z t Reactance of load CANCELS internal reactance of twoterminal circuit Second case: Load is pure resistance Load impedance for max. power transfer is Z load Rload Z t ELEC 308 20 Maximum Power Transfer Determine the maximum power that can be delivered to a load by the two-terminal resistance below if The load can be any complex impedance The load must be a pure resistance ELEC 308 21 Summary AC Power Real Power Reactive Power Apparent Power Power Factor Thevenin and Norton Maximum Power Transfer ELEC 308 22