14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary 14: Power in AC Circuits E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 1 / 11 Average Power 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 2 / 11 Average Power 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction 2 • Ideal Transformer v (t) • Transformer Applications Intantaneous Power dissipated in R: p(t) = R • Summary E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 2 / 11 Average Power 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction 2 • Ideal Transformer v (t) • Transformer Applications Intantaneous Power dissipated in R: p(t) = R • Summary E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 2 / 11 Average Power 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction 2 • Ideal Transformer v (t) • Transformer Applications Intantaneous Power dissipated in R: p(t) = R • Summary Average Power dissipated in R: T P = 1 p(t)dt T R0 E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 2 / 11 Average Power 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction 2 • Ideal Transformer v (t) • Transformer Applications Intantaneous Power dissipated in R: p(t) = R • Summary Average Power dissipated in R: T T P = 1 p(t)dt = 1 1 v2(t)dt T R0 R × T R0 E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 2 / 11 Average Power 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction 2 • Ideal Transformer v (t) • Transformer Applications Intantaneous Power dissipated in R: p(t) = R • Summary Average Power dissipated in R: 2 T T v (t) P = 1 p(t)dt = 1 1 v2(t)dt = h i T R0 R × T R0 R E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 2 / 11 Average Power 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction 2 • Ideal Transformer v (t) • Transformer Applications Intantaneous Power dissipated in R: p(t) = R • Summary Average Power dissipated in R: 2 T T v (t) P = 1 p(t)dt = 1 1 v2(t)dt = h i T R0 R × T R0 R v2(t) is the value of v2(t) averaged over time E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 2 / 11 Average Power 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction 2 • Ideal Transformer v (t) • Transformer Applications Intantaneous Power dissipated in R: p(t) = R • Summary Average Power dissipated in R: 2 T T v (t) P = 1 p(t)dt = 1 1 v2(t)dt = h i T R0 R × T R0 R v2(t) is the value of v2(t) averaged over time 2 We define the RMS Voltage (Root Mean Square): Vrms , v (t) ph i E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 2 / 11 Average Power 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction 2 • Ideal Transformer v (t) • Transformer Applications Intantaneous Power dissipated in R: p(t) = R • Summary Average Power dissipated in R: 2 T T v (t) P = 1 p(t)dt = 1 1 v2(t)dt = h i T R0 R × T R0 R v2(t) is the value of v2(t) averaged over time 2 We define the RMS Voltage (Root Mean Square): Vrms , v (t) ph i 2 2 v (t) (Vrms) The average power dissipated in R is P = h R i = R Vrms is the DC voltage that would cause R to dissipate the same power. E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 2 / 11 Average Power 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction 2 • Ideal Transformer v (t) • Transformer Applications Intantaneous Power dissipated in R: p(t) = R • Summary Average Power dissipated in R: 2 T T v (t) P = 1 p(t)dt = 1 1 v2(t)dt = h i T R0 R × T R0 R v2(t) is the value of v2(t) averaged over time 2 We define the RMS Voltage (Root Mean Square): Vrms , v (t) ph i 2 2 v (t) (Vrms) The average power dissipated in R is P = h R i = R Vrms is the DC voltage that would cause R to dissipate the same power. We use small letters for time-varying voltages and capital letters for time-invariant values. E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 2 / 11 Cosine Wave RMS 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. • Summary E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 3 / 11 Cosine Wave RMS 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. • Summary Squared Voltage: v2(t) = V 2 cos2 ωt = V 2 1 + 1 cos 2ωt 2 2 E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 3 / 11 Cosine Wave RMS 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. • Summary Squared Voltage: v2(t) = V 2 cos2 ωt = V 2 1 + 1 cos 2ωt 2 2 2 Mean Square Voltage: v2 = V since cos 2ωt averages to zero. 2 E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 3 / 11 Cosine Wave RMS 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. • Summary Squared Voltage: v2(t) = V 2 cos2 ωt = V 2 1 + 1 cos 2ωt 2 2 2 Mean Square Voltage: v2 = V since cos 2ωt averages to zero. 2 E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 3 / 11 Cosine Wave RMS 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. • Summary Squared Voltage: v2(t) = V 2 cos2 ωt = V 2 1 + 1 cos 2ωt 2 2 2 Mean Square Voltage: v2 = V since cos 2ωt averages to zero. 2 2 1 RMS Voltage: Vrms = v = V = 3.54 V ph i √2 E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 3 / 11 Cosine Wave RMS 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. • Summary Squared Voltage: v2(t) = V 2 cos2 ωt = V 2 1 + 1 cos 2ωt 2 2 2 Mean Square Voltage: v2 = V since cos 2ωt averages to zero. 2 2 1 RMS Voltage: Vrms = v = V = 3.54 V ph i √2 Note: Power engineers always use RMS voltages and currents exclusively and omit the “rms” subscript. For example UK Mains voltage = 230 V rms = 325 V peak. E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 3 / 11 Cosine Wave RMS 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. • Summary Squared Voltage: v2(t) = V 2 cos2 ωt = V 2 1 + 1 cos 2ωt 2 2 2 Mean Square Voltage: v2 = V since cos 2ωt averages to zero. 2 2 1 RMS Voltage: Vrms = v = V = 3.54 V = V ph i √2 e Note: Power engineers always use RMS voltages and currents exclusively and omit the “rms” subscript. For example UK Mains voltage = 230 V rms = 325 V peak. In this lecture course only, a ~ overbar means √2: thus V = 1 V . ÷ √2 e E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 3 / 11 Power Factor + 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer Suppose voltage and current phasors are: • Transformer Applications jθV • Summary V = V e | | I = I ejθI | | E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 4 / 11 Power Factor + 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer Suppose voltage and current phasors are: • Transformer Applications jθV • Summary V = V e v(t) = V cos(ωt + θV ) | | ⇔ | | jθI I = I e i(t) = I cos(ωt + θI ) | | ⇔ | | E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 4 / 11 Power Factor + 14: Power in AC Circuits • Average Power • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor