Slide 1Instantaneous Power: p(t)Average Power (P)Average Power: Special CasesAverage Power SummaryEffective or RMS ValuesWhy RMS Values?RMS in Everyday LifeClass Examples• Instantaneous power• Average power• RMS• ExamplesLecture 20. AC Power Analysis12Instantaneous Power: p(t)For AC circuits, the voltage and current arev(t) = VM cos(t+v)i(t) = IM cos(t+i)The instantaneous power is simply their productp(t) = v(t) i(t) = VM IM cos(t+v) cos(t+i)= ½VM IM [cos(v- i) + cos(2t+v +i)]ConstantTermWave of Twice Original Frequency3Average Power (P)•Calculate average power (integrate power over one cycle and divide by period)•Recall that passive sign convention says:P > 0, power is being absorbedP < 0, power is being supplied ivMMiMvMTttTtt- I V = dt + t I t V T = dt p(t)T = Pcos21coscos1100004Average Power: Special Cases•Purely resistive circuit: P = ½ VM IMThe power dissipated in a resistor is •Purely reactive circuit: P = 0–Capacitors and inductors are lossless elements and absorb no average power–A purely reactive network operates in a mode in which it stores energy over one part of the period and releases it over another partRI = R V = IV = PMMMM22212215Average Power SummaryCircuit Element Average PowerV or I sourceP = ½ VM IM cos(v- i)Resistor P = ½ VM IM = ½ IM2 RCapacitor orInductorP = 0Does the expression for the resistor power look identical to that for DC circuits?6Effective or RMS Values•Root-mean-square value (formula reads like the name: rms)•For a sinusoid: Irms = IM/2–For example, AC household outlets are around 120 Volts-rmsTttrmsTttrmsdttvTVanddttiTI0000)(1)(1227Why RMS Values?•The effective/rms current allows us to write average power expressions like those used in dc circuits (i.e., P=I²R), and that relation is really the basis for defining the rms value•The average power (P) is RIRVIVIVPIVIVPrmsrmsrmsrmsMMresistorivrmsrmsivMMsource2221coscos218RMS in Everyday Life•When we buy consumer electronics, the faceplate specifications provide the rms voltage and current values•For example, what is the rms current for a 1200 Watt hairdryer (although there is a small fan in a hairdryer, most of the power goes to a resistive heating element)?•What happens when two hairdryers are turned on at the same time in the bathroom?•How can I determine which uses more electricity---a plasma or an LCD HDTV?9Class Examples•Drill Problems P8-10, P8-11,
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