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Chapter-7

# Chapter-7 - Univ of Ontario Institute of Technology Faculty...

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Univ. of Ontario Institute of Technology Faculty of Engineering and Applied Science ' \$ % AC Power We have learned so far that in an electric circuit that is excited by a sinusoidal source, all the voltages and currents in the circuit are also sinusoidals of the same frequency. Consider the following circuit. Electric Circuits (ENGR2790U) 217 of 310

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AC Power ' \$ % Electric Circuits (ENGR2790U) 218 of 310
AC Power ' \$ % The voltage and current delivered to an arbitrary load can be written in the general forms: v ( t ) = V cos( ωt ) i ( t ) = I cos( ωt - θ I ) where V / I is the peak amplitudes of the sinusoidal voltage/current, and θ I is the phase delay of current with respect to the voltage. Then the instantaneous power dissipated by the load is given by p ( t ) = v ( t ) i ( t ) = V I cos( ωt ) cos( ωt - θ ) Electric Circuits (ENGR2790U) 219 of 310

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AC Power ' \$ % Using trigonometric identities, we can write the instantaneous power as p ( t ) = V I 2 cos( θ ) | {z } Constant + V I 2 cos(2 ωt - θ ) | {z } time-varying As can be seen, the instantaneous power dissipated by the load has an average component and a sinusoidal component. Electric Circuits (ENGR2790U) 220 of 310
AC Power ' \$ % Electric Circuits (ENGR2790U) 221 of 310

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AC Power ' \$ % Electric Circuits (ENGR2790U) 222 of 310
AC Power ' \$ % The average power P av is obtained as P av = 1 T Z T 0 p ( t ) dt = 1 T Z T 0 V I 2 cos θdt + 1 T Z T 0 V I 2 cos(2 ωt - θ ) dt = V I 2 cos θ (3) Note that Z ( ) = V ( ) I ( ) = V e j 0 Ie - = V I e = | Z | e Electric Circuits (ENGR2790U) 223 of 310

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AC Power ' \$ % Then, we can the average power as P av = 1 2 V 2 | Z | cos θ = 1 2 I 2 | Z | cos θ In North America, AC frequency is ω = 2 π 60 = 377 rad / s In Europe ω = 2 π 50 = 314 rad / s Electric Circuits (ENGR2790U) 224 of 310
AC Power ' \$ % Using rms values V rms = V 2 = ˜ V I rms = I 2 = ˜ I we can write the average power as P av = 1 2 V 2 | Z | cos θ = ˜ V 2 | Z | cos θ = 1 2 I 2 | Z | cos θ = ˜ I 2 | Z | cos θ = ˜ V ˜ I cos θ (4) Electric Circuits (ENGR2790U) 225 of 310

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AC Power ' \$ % We denote the rms phasors as ˜ V = V rms e V = ˜ V e V = ˜ V θ V ˜ I = I rms e I = ˜ Ie I = ˜ I θ I Example 7.1: Electric Circuits (ENGR2790U) 226 of 310
AC Power ' \$ % Example 7.2: Electric Circuits (ENGR2790U) 227 of 310

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Univ. of Ontario Institute of Technology Faculty of Engineering and Applied Science ' \$ % Complex Power Complex power is deﬁned a S = ˜ V ˜ I * Not that S = ˜ V ˜ I cos θ | {z
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Chapter-7 - Univ of Ontario Institute of Technology Faculty...

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