10.6: AC Power
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When a current I flows through a resistance R, the rate of dissipation of electrical energy as heat is IR2. If an alternating potential difference V=ˆVsinωt is applied across a resistance, then an alternating current I=ˆIsinωt will flow through it, and the rate at which energy is dissipated as heat will also change periodically. Of interest is the average rate of dissipation of electrical energy as heat during a complete cycle of period P=2π/ω.
Let W = instantaneous rate of dissipation of energy, and ¯W = average rate over a cycle of period P=2π/ω. Then
¯WP=∫P0Wdt=R∫P0I2dt=RˆI2∫P0sin2ωtdt=12RˆI2∫P0(1−cos2ωt)dt=12RˆI2[t−12ωsin2ωt]P=2π/ω0=12RˆI2P.
Thus
¯W=12RˆI2
The expression 12ˆI2 is the mean value of I2 over a complete cycle. Its square root ˆI/√2=0.707ˆI is the root mean square value of the current, IRMS. Thus the average rate of dissipation of electrical energy is
¯W=RI2RMS.
Likewise, the RMS EMF (pardon all the abbreviations) over a complete cycle is ˆV/√2.
Often when an AC current or voltage is quoted, it is the RMS value that is meant rather than the peak value. I recommend that in writing or conversation you always make it explicitly clear which you mean.
Also of interest is the mean induced voltage ¯V over half a cycle. (Over a full cycle, the mean voltage is, of course, zero.) We have
¯VP/2=∫P/20Vdt=ˆV∫P/20sinωtdt=ˆVω[cosωt]0P2=π2=ˆVω(1−cosπ)=2ˆVω.
Remembering that P=2π/ω, we see that
¯V=2ˆVπ=0.6366ˆV=2√2VRMSπ=0.9003VRMS.