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 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 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 ) | | ⇔ | | Power dissipated in load Z is p(t) = v(t)i(t) = V I cos(ωt + θV )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 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 ) | | ⇔ | | Power dissipated in load Z is p(t) = v(t)i(t) = V I cos(ωt + θV )cos(ωt + θI ) 1 | || | 1 = V I cos (2ωt + θV + θI ) + cos(θV θI ) | || | 2 2 −
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 ) | | ⇔ | | Power dissipated in load Z is p(t) = v(t)i(t) = V I cos(ωt + θV )cos(ωt + θI ) 1 | || | 1 = V I 2 cos (2ωt + θV + θI ) + 2 cos(θV θI ) |1 || | 1 − = V I cos(θV θI ) + V I cos (2ωt + θV + θI ) 2 | || | − 2 | || |
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 ) | | ⇔ | | Power dissipated in load Z is p(t) = v(t)i(t) = V I cos(ωt + θV )cos(ωt + θI ) 1 | || | 1 = V I 2 cos (2ωt + θV + θI ) + 2 cos(θV θI ) |1 || | 1 − = V I cos(θV θI ) + V I cos (2ωt + θV + θI ) 2 | || | − 2 | || |
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 ) | | ⇔ | | Power dissipated in load Z is p(t) = v(t)i(t) = V I cos(ωt + θV )cos(ωt + θI ) 1 | || | 1 = V I 2 cos (2ωt + θV + θI ) + 2 cos(θV θI ) |1 || | 1 − = V I cos(θV θI ) + V I cos (2ωt + θV + θI ) 2 | || | − 2 | || | 1 Average power: P = V I cos(φ) where φ = θV θI 2 | || | −
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 ) | | ⇔ | | Power dissipated in load Z is p(t) = v(t)i(t) = V I cos(ωt + θV )cos(ωt + θI ) 1 | || | 1 = V I 2 cos (2ωt + θV + θI ) + 2 cos(θV θI ) |1 || | 1 − = V I cos(θV θI ) + V I cos (2ωt + θV + θI ) 2 | || | − 2 | || | 1 Average power: P = V I cos(φ) where φ = θV θI 2 | || | − = V I cos(φ)
e e
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 ) | | ⇔ | | Power dissipated in load Z is p(t) = v(t)i(t) = V I cos(ωt + θV )cos(ωt + θI ) 1 | || | 1 = V I 2 cos (2ωt + θV + θI ) + 2 cos(θV θI ) |1 || | 1 − = V I cos(θV θI ) + V I cos (2ωt + θV + θI ) 2 | || | − 2 | || | 1 Average power: P = V I cos(φ) where φ = θV θI 2 | || | − = V I cos(φ) cos φ is the power factor
e e
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 ) | | ⇔ | | Power dissipated in load Z is p(t) = v(t)i(t) = V I cos(ωt + θV )cos(ωt + θI ) 1 | || | 1 = V I 2 cos (2ωt + θV + θI ) + 2 cos(θV θI ) |1 || | 1 − = V I cos(θV θI ) + V I cos (2ωt + θV + θI ) 2 | || | − 2 | || | 1 Average power: P = V I cos(φ) where φ = θV θI 2 | || | − = V I cos(φ) cos φ is the power factor e e φ > 0 a lagging power factor (normal case: Current lags Voltage) ⇔ φ < 0 a leading power factor (rare case: Current leads Voltage) ⇔ E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 4 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI Ideal Transformer V I∗ = V e I e− • Transformer Applications × × • Summary e e e e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI j(θV θI ) Ideal Transformer V I∗ = V e I e− = V I e − • Transformer Applications × × • Summary e e e e e e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI j(θV θI ) Ideal Transformer V I∗ = V e I e− = V I e − • Transformer Applications × × • Summary e e e e e e jφ = V I e
e e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI j(θV θI ) Ideal Transformer V I∗ = V e I e− = V I e − • Transformer Applications × × • Summary e e e e e e jφ = V I e = V I cos φ + j V I sin φ
e e e e e e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI j(θV θI ) Ideal Transformer V I∗ = V e I e− = V I e − • Transformer Applications × × • Summary e e e e e e jφ = V I e = V I cos φ + j V I sin φ e e e e e e = P + jQ
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI j(θV θI ) Ideal Transformer V I∗ = V e I e− = V I e − • Transformer Applications × × • Summary e e e e e e jφ = V I e = V I cos φ + j V I sin φ e e e e e e = P + jQ
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI j(θV θI ) Ideal Transformer V I∗ = V e I e− = V I e − • Transformer Applications × × • Summary e e e e e e jφ = V I e = V I cos φ + j V I sin φ e e e e e e = P + jQ
Complex Power: S , V I∗ = P + jQ measured in Volt-Amps (VA) e e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI j(θV θI ) Ideal Transformer V I∗ = V e I e− = V I e − • Transformer Applications × × • Summary e e e e e e jφ = V I e = V I cos φ + j V I sin φ e e e e e e = P + jQ
Complex Power: S , V I∗ = P + jQ measured in Volt-Amps (VA) , e e Apparent Power: S V I measured in Volt-Amps (VA) | | e e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI j(θV θI ) Ideal Transformer V I∗ = V e I e− = V I e − • Transformer Applications × × • Summary e e e e e e jφ = V I e = V I cos φ + j V I sin φ e e e e e e = P + jQ
Complex Power: S , V I∗ = P + jQ measured in Volt-Amps (VA) , e e Apparent Power: S V I measured in Volt-Amps (VA) | | Average Power: P , e( S )e measured in Watts (W) ℜ
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI j(θV θI ) Ideal Transformer V I∗ = V e I e− = V I e − • Transformer Applications × × • Summary e e e e e e jφ = V I e = V I cos φ + j V I sin φ e e e e e e = P + jQ
Complex Power: S , V I∗ = P + jQ measured in Volt-Amps (VA) , e e Apparent Power: S V I measured in Volt-Amps (VA) | | Average Power: P , e( S )e measured in Watts (W) ℜ Reactive Power: Q , (S) Measured in Volt-Amps Reactive (VAR) ℑ
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI j(θV θI ) Ideal Transformer V I∗ = V e I e− = V I e − • Transformer Applications × × • Summary e e e e e e jφ = V I e = V I cos φ + j V I sin φ e e e e e e = P + jQ
Complex Power: S , V I∗ = P + jQ measured in Volt-Amps (VA) , e e Apparent Power: S V I measured in Volt-Amps (VA) | | Average Power: P , e( S )e measured in Watts (W) ℜ Reactive Power: Q , (S) Measured in Volt-Amps Reactive (VAR) ℑ Power Factor: cos φ , cos ∠V ∠I = P − S e e | |
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI j(θV θI ) Ideal Transformer V I∗ = V e I e− = V I e − • Transformer Applications × × • Summary e e e e e e jφ = V I e = V I cos φ + j V I sin φ e e e e e e = P + jQ
Complex Power: S , V I∗ = P + jQ measured in Volt-Amps (VA) , e e Apparent Power: S V I measured in Volt-Amps (VA) | | Average Power: P , e( S )e measured in Watts (W) ℜ Reactive Power: Q , (S) Measured in Volt-Amps Reactive (VAR) ℑ Power Factor: cos φ , cos ∠V ∠I = P − S e e | | Machines and transformers have capacity limits and power losses that are independent of cos φ; their ratings are always given in apparent power.
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 5 / 11 Complex Power
14: Power in AC Circuits 1 jθV 1 jθI • Average Power If V = V e and I˜ = I e √2 √2 • Cosine Wave RMS | | | | • e Power Factor + , • Complex Power The complex power absorbed by Z is S V I∗ • Power in R, L, C where * means complex conjugate. × • Tellegen’s Theorem e e • Power Factor Correction • jθV jθI j(θV θI ) Ideal Transformer V I∗ = V e I e− = V I e − • Transformer Applications × × • Summary e e e e e e jφ = V I e = V I cos φ + j V I sin φ e e e e e e = P + jQ
Complex Power: S , V I∗ = P + jQ measured in Volt-Amps (VA) , e e Apparent Power: S V I measured in Volt-Amps (VA) | | Average Power: P , e( S )e measured in Watts (W) ℜ Reactive Power: Q , (S) Measured in Volt-Amps Reactive (VAR) ℑ Power Factor: cos φ , cos ∠V ∠I = P − S e e | | Machines and transformers have capacity limits and power losses that are independent of cos φ; their ratings are always given in apparent power. Power Company: Costs apparent power, Revenue average power. E1.1 Analysis of Circuits (2017-10213) ∝ ∝ AC Power: 14 – 5 / 11 Power in R, L, C
14: Power in AC Circuits • Average Power For any impedance, Z, complex power absorbed: S = V I∗ = P + jQ • Cosine Wave RMS • Power Factor + e e • 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 – 6 / 11 Power in R, L, C
14: Power in AC Circuits • Average Power For any impedance, Z, complex power absorbed: S = V I∗ = P + jQ 2 • Cosine Wave RMS 2 2 Ve • e e Power Factor + Using (a) V = IZ (b) I I∗ = I we get S = I Z = ∗ • |Z| Complex Power × • Power in R, L, C e e e e e e • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 6 / 11 Power in R, L, C
14: Power in AC Circuits • Average Power For any impedance, Z, complex power absorbed: S = V I∗ = P + jQ 2 • Cosine Wave RMS 2 2 Ve • e e Power Factor + Using (a) V = IZ (b) I I∗ = I we get S = I Z = ∗ • |Z| Complex Power × • Power in R, L, C e e e e e e 2 • Tellegen’s Theorem 2 Ve • Power Factor Correction Resistor: S = I R = | | φ = 0 • Ideal Transformer R
• Transformer Applications e • Summary Absorbs average power, no VARs (Q = 0)
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 6 / 11 Power in R, L, C
14: Power in AC Circuits • Average Power For any impedance, Z, complex power absorbed: S = V I∗ = P + jQ 2 • Cosine Wave RMS 2 2 Ve • e e Power Factor + Using (a) V = IZ (b) I I∗ = I we get S = I Z = ∗ • |Z| Complex Power × • Power in R, L, C e e e e e e 2 • Tellegen’s Theorem 2 Ve • Power Factor Correction Resistor: S = I R = | | φ = 0 • Ideal Transformer R
• Transformer Applications e • Summary Absorbs average power, no VARs (Q = 0)
2 2 Ve Inductor: S j I ωL j φ = = |ωL| = +90◦ e No average power, Absorbs VARs (Q > 0)
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 6 / 11 Power in R, L, C
14: Power in AC Circuits • Average Power For any impedance, Z, complex power absorbed: S = V I∗ = P + jQ 2 • Cosine Wave RMS 2 2 Ve • e e Power Factor + Using (a) V = IZ (b) I I∗ = I we get S = I Z = ∗ • |Z| Complex Power × • Power in R, L, C e e e e e e 2 • Tellegen’s Theorem 2 Ve • Power Factor Correction Resistor: S = I R = | | φ = 0 • Ideal Transformer R
• Transformer Applications e • Summary Absorbs average power, no VARs (Q = 0)
2 2 Ve Inductor: S j I ωL j φ = = |ωL| = +90◦ e No average power, Absorbs VARs (Q > 0)
2 Ie 2 Capacitor: S = j | | = j V ωC φ = 90◦ − ωC − − e No average power, Generates VARs (Q < 0)
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 6 / 11 Power in R, L, C
14: Power in AC Circuits • Average Power For any impedance, Z, complex power absorbed: S = V I∗ = P + jQ 2 • Cosine Wave RMS 2 2 Ve • e e Power Factor + Using (a) V = IZ (b) I I∗ = I we get S = I Z = ∗ • |Z| Complex Power × • Power in R, L, C e e e e e e 2 • Tellegen’s Theorem 2 Ve • Power Factor Correction Resistor: S = I R = | | φ = 0 • Ideal Transformer R
• Transformer Applications e • Summary Absorbs average power, no VARs (Q = 0)
2 2 Ve Inductor: S j I ωL j φ = = |ωL| = +90◦ e No average power, Absorbs VARs (Q > 0)
2 Ie 2 Capacitor: S = j | | = j V ωC φ = 90◦ − ωC − − e No average power, Generates VARs (Q < 0)
VARs are generated by capacitors and absorbed by inductors
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 6 / 11 Power in R, L, C
14: Power in AC Circuits • Average Power For any impedance, Z, complex power absorbed: S = V I∗ = P + jQ 2 • Cosine Wave RMS 2 2 Ve • e e Power Factor + Using (a) V = IZ (b) I I∗ = I we get S = I Z = ∗ • |Z| Complex Power × • Power in R, L, C e e e e e e 2 • Tellegen’s Theorem 2 Ve • Power Factor Correction Resistor: S = I R = | | φ = 0 • Ideal Transformer R
• Transformer Applications e • Summary Absorbs average power, no VARs (Q = 0)
2 2 Ve Inductor: S j I ωL j φ = = |ωL| = +90◦ e No average power, Absorbs VARs (Q > 0)
2 Ie 2 Capacitor: S = j | | = j V ωC φ = 90◦ − ωC − − e No average power, Generates VARs (Q < 0)
VARs are generated by capacitors and absorbed by inductors The phase, φ, of the absorbed power, S, equals the phase of Z
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 6 / 11 Tellegen's Theorem
14: Power in AC Circuits Tellegen’s Theorem: The complex power, S, dissipated in any circuit’s • Average Power • Cosine Wave RMS components sums to zero. • Power Factor + • Complex Power x voltage at node n • Power in R, L, C n = • Tellegen’s Theorem Vb, Ib = voltage/current in branch b • Power Factor Correction • Ideal Transformer (obeying passive sign convention) • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 7 / 11 Tellegen's Theorem
14: Power in AC Circuits Tellegen’s Theorem: The complex power, S, dissipated in any circuit’s • Average Power • Cosine Wave RMS components sums to zero. • Power Factor + • Complex Power x voltage at node n • Power in R, L, C n = • Tellegen’s Theorem Vb, Ib = voltage/current in branch b • Power Factor Correction • Ideal Transformer (obeying passive sign convention) • Transformer Applications • Summary 1 if V starts from node n b , − abn +1 if Vb ends at node n 0 else
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 7 / 11 Tellegen's Theorem
14: Power in AC Circuits Tellegen’s Theorem: The complex power, S, dissipated in any circuit’s • Average Power • Cosine Wave RMS components sums to zero. • Power Factor + • Complex Power x voltage at node n • Power in R, L, C n = • Tellegen’s Theorem Vb, Ib = voltage/current in branch b • Power Factor Correction • Ideal Transformer (obeying passive sign convention) • Transformer Applications • Summary 1 if V starts from node n b , − abn +1 if Vb ends at node n 0 else e.g. branch 4 goes from 2 to 3 a4 = [0, 1, 1] ⇒ ∗ −
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 7 / 11 Tellegen's Theorem
14: Power in AC Circuits Tellegen’s Theorem: The complex power, S, dissipated in any circuit’s • Average Power • Cosine Wave RMS components sums to zero. • Power Factor + • Complex Power x voltage at node n • Power in R, L, C n = • Tellegen’s Theorem Vb, Ib = voltage/current in branch b • Power Factor Correction • Ideal Transformer (obeying passive sign convention) • Transformer Applications • Summary 1 if V starts from node n b , − abn +1 if Vb ends at node n 0 else e.g. branch 4 goes from 2 to 3 a4 = [0, 1, 1] ⇒ ∗ − Branch voltages: Vb = abnxn (e.g. V4 = x3 x2) Pn −
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 7 / 11 Tellegen's Theorem
14: Power in AC Circuits Tellegen’s Theorem: The complex power, S, dissipated in any circuit’s • Average Power • Cosine Wave RMS components sums to zero. • Power Factor + • Complex Power x voltage at node n • Power in R, L, C n = • Tellegen’s Theorem Vb, Ib = voltage/current in branch b • Power Factor Correction • Ideal Transformer (obeying passive sign convention) • Transformer Applications • Summary 1 if V starts from node n b , − abn +1 if Vb ends at node n 0 else e.g. branch 4 goes from 2 to 3 a4 = [0, 1, 1] ⇒ ∗ − Branch voltages: Vb = abnxn (e.g. V4 = x3 x2) Pn − KCL @ node n: abnIb = 0 abnI∗ = 0 Pb ⇒ Pb b
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 7 / 11 Tellegen's Theorem
14: Power in AC Circuits Tellegen’s Theorem: The complex power, S, dissipated in any circuit’s • Average Power • Cosine Wave RMS components sums to zero. • Power Factor + • Complex Power x voltage at node n • Power in R, L, C n = • Tellegen’s Theorem Vb, Ib = voltage/current in branch b • Power Factor Correction • Ideal Transformer (obeying passive sign convention) • Transformer Applications • Summary 1 if V starts from node n b , − abn +1 if Vb ends at node n 0 else e.g. branch 4 goes from 2 to 3 a4 = [0, 1, 1] ⇒ ∗ − Branch voltages: Vb = abnxn (e.g. V4 = x3 x2) Pn − KCL @ node n: abnIb = 0 abnI∗ = 0 Pb ⇒ Pb b Tellegen: VbI∗ = abnxnI∗ Pb b Pb Pn b
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 7 / 11 Tellegen's Theorem
14: Power in AC Circuits Tellegen’s Theorem: The complex power, S, dissipated in any circuit’s • Average Power • Cosine Wave RMS components sums to zero. • Power Factor + • Complex Power x voltage at node n • Power in R, L, C n = • Tellegen’s Theorem Vb, Ib = voltage/current in branch b • Power Factor Correction • Ideal Transformer (obeying passive sign convention) • Transformer Applications • Summary 1 if V starts from node n b , − abn +1 if Vb ends at node n 0 else e.g. branch 4 goes from 2 to 3 a4 = [0, 1, 1] ⇒ ∗ − Branch voltages: Vb = abnxn (e.g. V4 = x3 x2) Pn − KCL @ node n: abnIb = 0 abnI∗ = 0 Pb ⇒ Pb b Tellegen: VbI∗ = abnxnI∗ Pb b Pb Pn b
= abnI∗xn Pn Pb b
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 7 / 11 Tellegen's Theorem
14: Power in AC Circuits Tellegen’s Theorem: The complex power, S, dissipated in any circuit’s • Average Power • Cosine Wave RMS components sums to zero. • Power Factor + • Complex Power x voltage at node n • Power in R, L, C n = • Tellegen’s Theorem Vb, Ib = voltage/current in branch b • Power Factor Correction • Ideal Transformer (obeying passive sign convention) • Transformer Applications • Summary 1 if V starts from node n b , − abn +1 if Vb ends at node n 0 else e.g. branch 4 goes from 2 to 3 a4 = [0, 1, 1] ⇒ ∗ − Branch voltages: Vb = abnxn (e.g. V4 = x3 x2) Pn − KCL @ node n: abnIb = 0 abnI∗ = 0 Pb ⇒ Pb b Tellegen: VbI∗ = abnxnI∗ Pb b Pb Pn b
= abnI∗xn= xn abnI∗ Pn Pb b Pn Pb b
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 7 / 11 Tellegen's Theorem
14: Power in AC Circuits Tellegen’s Theorem: The complex power, S, dissipated in any circuit’s • Average Power • Cosine Wave RMS components sums to zero. • Power Factor + • Complex Power x voltage at node n • Power in R, L, C n = • Tellegen’s Theorem Vb, Ib = voltage/current in branch b • Power Factor Correction • Ideal Transformer (obeying passive sign convention) • Transformer Applications • Summary 1 if V starts from node n b , − abn +1 if Vb ends at node n 0 else e.g. branch 4 goes from 2 to 3 a4 = [0, 1, 1] ⇒ ∗ − Branch voltages: Vb = abnxn (e.g. V4 = x3 x2) Pn − KCL @ node n: abnIb = 0 abnI∗ = 0 Pb ⇒ Pb b Tellegen: VbI∗ = abnxnI∗ Pb b Pb Pn b
= abnI∗xn= xn abnI∗ = xn 0 Pn Pb b Pn Pb b Pn ×
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 7 / 11 Tellegen's Theorem
14: Power in AC Circuits Tellegen’s Theorem: The complex power, S, dissipated in any circuit’s • Average Power • Cosine Wave RMS components sums to zero. • Power Factor + • Complex Power x voltage at node n • Power in R, L, C n = • Tellegen’s Theorem Vb, Ib = voltage/current in branch b • Power Factor Correction • Ideal Transformer (obeying passive sign convention) • Transformer Applications • Summary 1 if V starts from node n b , − abn +1 if Vb ends at node n 0 else e.g. branch 4 goes from 2 to 3 a4 = [0, 1, 1] ⇒ ∗ − Branch voltages: Vb = abnxn (e.g. V4 = x3 x2) Pn − KCL @ node n: abnIb = 0 abnI∗ = 0 Pb ⇒ Pb b Tellegen: VbI∗ = abnxnI∗ Pb b Pb Pn b
= abnI∗xn= xn abnI∗ = xn 0 Pn Pb b Pn Pb b Pn × Note: Sb = 0 Pb = 0 and also Qb = 0. Pb ⇒ Pb Pb E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 7 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie = R + ZL • Complex Power • Power in R, L, C e • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A = R + ZL = 46 32 9 • Complex Power − • Power in R, L, C e • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − • Power in R, L, C e • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA • Tellegen’s Theorem • Power Factor Correction e e • Ideal Transformer • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem • Power Factor Correction e e • Ideal Transformer • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications Add parallel capacitor of 300 µF: • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications • Add parallel capacitor of 300 µF: Summary 1 ZC = = 10.6j Ω jωC −
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications • Add parallel capacitor of 300 µF: Summary 1 ZC = = 10.6j Ω IC = 21.7j A jωC − ⇒ e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications • Add parallel capacitor of 300 µF: Summary 1 ZC = = 10.6j Ω IC = 21.7j A jωC − ⇒ I = 46 j11.2 A = 47∠ e14◦ A − − e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications • Add parallel capacitor of 300 µF: Summary 1 ZC = = 10.6j Ω IC = 21.7j A jωC − ⇒ I = 46 j11.2 A = 47∠ e14◦ A − − e
SC = V I∗ = j5 kVA C − e e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications • Add parallel capacitor of 300 µF: Summary 1 ZC = = 10.6j Ω IC = 21.7j A jωC − ⇒ I = 46 j11.2 A = 47∠ e14◦ A − − e
SC = V I∗ = j5 kVA C − S = V eI∗e= 10.6 + j2.6 kVA e e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications • Add parallel capacitor of 300 µF: Summary 1 ZC = = 10.6j Ω IC = 21.7j A jωC − ⇒ I = 46 j11.2 A = 47∠ e14◦ A − − e
SC = V I∗ = j5 kVA C − S = V eI∗e= 10.6 + j2.6 kVA = 10.9∠14◦ kVA e e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications • Add parallel capacitor of 300 µF: Summary 1 ZC = = 10.6j Ω IC = 21.7j A jωC − ⇒ I = 46 j11.2 A = 47∠ e14◦ A − − e
SC = V I∗ = j5 kVA C − S = V eI∗e= 10.6 + j2.6 kVA = 10.9∠14◦ kVA P cos φe=e S = cos 14◦ = 0.97 | |
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications • Add parallel capacitor of 300 µF: Summary 1 ZC = = 10.6j Ω IC = 21.7j A jωC − ⇒ I = 46 j11.2 A = 47∠ e14◦ A − − e
SC = V I∗ = j5 kVA C − S = V eI∗e= 10.6 + j2.6 kVA = 10.9∠14◦ kVA P cos φe=e S = cos 14◦ = 0.97 | |
Average power to motor, P , is 10.6 kW in both cases.
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications • Add parallel capacitor of 300 µF: Summary 1 ZC = = 10.6j Ω IC = 21.7j A jωC − ⇒ I = 46 j11.2 A = 47∠ e14◦ A − − e
SC = V I∗ = j5 kVA C − S = V eI∗e= 10.6 + j2.6 kVA = 10.9∠14◦ kVA P cos φe=e S = cos 14◦ = 0.97 | |
Average power to motor, P , is 10.6 kW in both cases. I , reduced from 56.5 47 A( 16%) lower losses. ց − ⇒ e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications • Add parallel capacitor of 300 µF: Summary 1 ZC = = 10.6j Ω IC = 21.7j A jωC − ⇒ I = 46 j11.2 A = 47∠ e14◦ A − − e
SC = V I∗ = j5 kVA C − S = V eI∗e= 10.6 + j2.6 kVA = 10.9∠14◦ kVA P cos φe=e S = cos 14◦ = 0.97 | |
Average power to motor, P , is 10.6 kW in both cases. I , reduced from 56.5 47 A( 16%) lower losses. ց − ⇒ Effecte of C: VARs = 7.6 2.6 kVAR ց E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Power Factor Correction
14: Power in AC Circuits • Average Power V = 230. Motor modelled as 5 7j Ω. • Cosine Wave RMS Ve Ve || • Power Factor + Ie j . A . ∠ = R + ZL = 46 32 9 = 56 5 36◦ • Complex Power − − ∠ • Power in R, L, C Se = V I∗ = 10.6 + j7.6 kVA = 13 36◦ kVA • Tellegen’s Theorem cos φ = P = cos 36 = 0.81 • Power Factor Correction e e S ◦ • Ideal Transformer | | • Transformer Applications • Add parallel capacitor of 300 µF: Summary 1 ZC = = 10.6j Ω IC = 21.7j A jωC − ⇒ I = 46 j11.2 A = 47∠ e14◦ A − − e
SC = V I∗ = j5 kVA C − S = V eI∗e= 10.6 + j2.6 kVA = 10.9∠14◦ kVA P cos φe=e S = cos 14◦ = 0.97 | |
Average power to motor, P , is 10.6 kW in both cases. I , reduced from 56.5 47 A( 16%) lower losses. ց − ⇒ Effecte of C: VARs = 7.6 2.6 kVAR , power factor = 0.81 0.97. ց ր E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 8 / 11 Ideal Transformer
14: Power in AC Circuits A transformer has 2 windings on the same magnetic • Average Power ≥ • Cosine Wave RMS core. • 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 – 9 / 11 Ideal Transformer
14: Power in AC Circuits A transformer has 2 windings on the same magnetic • Average Power ≥ • Cosine Wave RMS core. • Power Factor + • Complex Power Ampère’s law: N I = lΦ ; Faraday’s law: Vr = dΦ . • Power in R, L, C r r µA Nr dt • Tellegen’s Theorem P • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 9 / 11 Ideal Transformer
14: Power in AC Circuits A transformer has 2 windings on the same magnetic • Average Power ≥ • Cosine Wave RMS core. • Power Factor + • Complex Power lΦ Vr dΦ Ampère’s law: NrIr = ; Faraday’s law: = . • Power in R, L, C P µA Nr dt • Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings. • Power Factor Correction • Ideal Transformer The indicates the voltage polarity of each winding. • Transformer Applications • • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 9 / 11 Ideal Transformer
14: Power in AC Circuits A transformer has 2 windings on the same magnetic • Average Power ≥ • Cosine Wave RMS core. • Power Factor + • Complex Power lΦ Vr dΦ Ampère’s law: NrIr = ; Faraday’s law: = . • Power in R, L, C P µA Nr dt • Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings. • Power Factor Correction • Ideal Transformer The indicates the voltage polarity of each winding. • Transformer Applications • V1 V2 V3 • Summary Since is the same for all windings, . Φ N1 = N2 = N3
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 9 / 11 Ideal Transformer
14: Power in AC Circuits A transformer has 2 windings on the same magnetic • Average Power ≥ • Cosine Wave RMS core. • Power Factor + • Complex Power lΦ Vr dΦ Ampère’s law: NrIr = ; Faraday’s law: = . • Power in R, L, C P µA Nr dt • Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings. • Power Factor Correction • Ideal Transformer The indicates the voltage polarity of each winding. • Transformer Applications • V1 V2 V3 • Summary Since is the same for all windings, . Φ N1 = N2 = N3 Assume µ N I + N I + N I = 0 → ∞ ⇒ 1 1 2 2 3 3
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 9 / 11 Ideal Transformer
14: Power in AC Circuits A transformer has 2 windings on the same magnetic • Average Power ≥ • Cosine Wave RMS core. • Power Factor + • Complex Power lΦ Vr dΦ Ampère’s law: NrIr = ; Faraday’s law: = . • Power in R, L, C P µA Nr dt • Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings. • Power Factor Correction • Ideal Transformer The indicates the voltage polarity of each winding. • Transformer Applications • V1 V2 V3 • Summary Since is the same for all windings, . Φ N1 = N2 = N3 Assume µ N I + N I + N I = 0 → ∞ ⇒ 1 1 2 2 3 3 These two equations allow you to solve circuits and also imply that Si = 0. P
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 9 / 11 Ideal Transformer
14: Power in AC Circuits A transformer has 2 windings on the same magnetic • Average Power ≥ • Cosine Wave RMS core. • Power Factor + • Complex Power lΦ Vr dΦ Ampère’s law: NrIr = ; Faraday’s law: = . • Power in R, L, C P µA Nr dt • Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings. • Power Factor Correction • Ideal Transformer The indicates the voltage polarity of each winding. • Transformer Applications • V1 V2 V3 • Summary Since is the same for all windings, . Φ N1 = N2 = N3 Assume µ N I + N I + N I = 0 → ∞ ⇒ 1 1 2 2 3 3 These two equations allow you to solve circuits and also imply that Si = 0. P Special Case:
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 9 / 11 Ideal Transformer
14: Power in AC Circuits A transformer has 2 windings on the same magnetic • Average Power ≥ • Cosine Wave RMS core. • Power Factor + • Complex Power lΦ Vr dΦ Ampère’s law: NrIr = ; Faraday’s law: = . • Power in R, L, C P µA Nr dt • Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings. • Power Factor Correction • Ideal Transformer The indicates the voltage polarity of each winding. • Transformer Applications • V1 V2 V3 • Summary Since is the same for all windings, . Φ N1 = N2 = N3 Assume µ N I + N I + N I = 0 → ∞ ⇒ 1 1 2 2 3 3 These two equations allow you to solve circuits and also imply that Si = 0. P Special Case:
For a 2-winding transformer this simplifies to N2 N1 V = V and IL = I = I 2 N1 1 − 2 N2 1
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 9 / 11 Ideal Transformer
14: Power in AC Circuits A transformer has 2 windings on the same magnetic • Average Power ≥ • Cosine Wave RMS core. • Power Factor + • Complex Power lΦ Vr dΦ Ampère’s law: NrIr = ; Faraday’s law: = . • Power in R, L, C P µA Nr dt • Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings. • Power Factor Correction • Ideal Transformer The indicates the voltage polarity of each winding. • Transformer Applications • V1 V2 V3 • Summary Since is the same for all windings, . Φ N1 = N2 = N3 Assume µ N I + N I + N I = 0 → ∞ ⇒ 1 1 2 2 3 3 These two equations allow you to solve circuits and also imply that Si = 0. P Special Case:
For a 2-winding transformer this simplifies to N2 N1 V = V and IL = I = I 2 N1 1 − 2 N2 1 2 2 Hence V1 = N1 V2 = N1 Z I1 N2 IL N2
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 9 / 11 Ideal Transformer
14: Power in AC Circuits A transformer has 2 windings on the same magnetic • Average Power ≥ • Cosine Wave RMS core. • Power Factor + • Complex Power lΦ Vr dΦ Ampère’s law: NrIr = ; Faraday’s law: = . • Power in R, L, C P µA Nr dt • Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings. • Power Factor Correction • Ideal Transformer The indicates the voltage polarity of each winding. • Transformer Applications • V1 V2 V3 • Summary Since is the same for all windings, . Φ N1 = N2 = N3 Assume µ N I + N I + N I = 0 → ∞ ⇒ 1 1 2 2 3 3 These two equations allow you to solve circuits and also imply that Si = 0. P Special Case:
For a 2-winding transformer this simplifies to N2 N1 V = V and IL = I = I 2 N1 1 − 2 N2 1 2 2 Hence V1 = N1 V2 = N1 Z I1 N2 IL N2 2 Equivalent to a reflected impedance of N1 Z N2 E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 9 / 11 Transformer Applications
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 – 10 / 11 Transformer Applications
14: Power in AC Circuits Power Transmission • Average Power • Cosine Wave RMS • Power Factor + Suppose a power transmission cable has 1Ω resistance. • Complex Power 100 kVA@ 1 kV = 100 A I2R = 10 kW losses. • Power in R, L, C ⇒ • Tellegen’s Theorem e • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 10 / 11 Transformer Applications
14: Power in AC Circuits Power Transmission • Average Power • Cosine Wave RMS • Power Factor + Suppose a power transmission cable has 1Ω resistance. • Complex Power 100 kVA@ 1 kV = 100 A I2R = 10 kW losses. • Power in R, L, C ⇒ 2 • Tellegen’s Theorem 100 kVA@ 100 kV = 1 A Ie R = 1 W losses. • Power Factor Correction ⇒ • Ideal Transformer e • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 10 / 11 Transformer Applications
14: Power in AC Circuits Power Transmission • Average Power • Cosine Wave RMS • Power Factor + Suppose a power transmission cable has 1Ω resistance. • Complex Power 100 kVA@ 1 kV = 100 A I2R = 10 kW losses. • Power in R, L, C ⇒ 2 • Tellegen’s Theorem 100 kVA@ 100 kV = 1 A Ie R = 1 W losses. • Power Factor Correction ⇒ • Ideal Transformer Voltage Conversion e • Transformer Applications • Summary Electronic equipment requires 20 V but mains voltage is 240 V . ≤ ∼
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 10 / 11 Transformer Applications
14: Power in AC Circuits Power Transmission • Average Power • Cosine Wave RMS • Power Factor + Suppose a power transmission cable has 1Ω resistance. • Complex Power 100 kVA@ 1 kV = 100 A I2R = 10 kW losses. • Power in R, L, C ⇒ 2 • Tellegen’s Theorem 100 kVA@ 100 kV = 1 A Ie R = 1 W losses. • Power Factor Correction ⇒ • Ideal Transformer Voltage Conversion e • Transformer Applications • Summary Electronic equipment requires 20 V but mains voltage is 240 V . ≤ ∼ Interference protection
Microphone on long cable is susceptible to interference from nearby mains cables. An N : 1 transformer reduces the microphone voltage by N but reduces interference by N 2.
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 10 / 11 Transformer Applications
14: Power in AC Circuits Power Transmission • Average Power • Cosine Wave RMS • Power Factor + Suppose a power transmission cable has 1Ω resistance. • Complex Power 100 kVA@ 1 kV = 100 A I2R = 10 kW losses. • Power in R, L, C ⇒ 2 • Tellegen’s Theorem 100 kVA@ 100 kV = 1 A Ie R = 1 W losses. • Power Factor Correction ⇒ • Ideal Transformer Voltage Conversion e • Transformer Applications • Summary Electronic equipment requires 20 V but mains voltage is 240 V . ≤ ∼ Interference protection
Microphone on long cable is susceptible to interference from nearby mains cables. An N : 1 transformer reduces the microphone voltage by N but reduces interference by N 2.
Isolation
There is no electrical connection between the windings of a transformer so circuitry (or people) on one side will not be endangered by a failure that results in high voltages on the other side.
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 10 / 11 Summary
14: Power in AC Circuits 1 • Average Power Complex Power: S = V I = P + jQ where V = V = V . ∗ rms √2 • Cosine Wave RMS • • Power Factor + e e e • 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 – 11 / 11 Summary
14: Power in AC Circuits 1 • Average Power Complex Power: where . S = V I∗ = P + jQ V = Vrms = √ V • 2 Cosine Wave RMS • 2 e 2 • Power Factor + e e V e ∗ • Complex Power For an impedance Z: S = I Z = | | ◦ Z • Power in R, L, C e • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 11 / 11 Summary
14: Power in AC Circuits 1 • Average Power Complex Power: where . S = V I∗ = P + jQ V = Vrms = √ V • 2 Cosine Wave RMS • 2 e 2 • Power Factor + e e V e ∗ • Complex Power For an impedance Z: S = I Z = | | ◦ Z • Power in R, L, C e • Tellegen’s Theorem Apparent Power: S = V I used for machine ratings. • Power Factor Correction ◦ | | • Ideal Transformer e e • Transformer Applications • Summary
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 11 / 11 Summary
14: Power in AC Circuits 1 • Average Power Complex Power: where . S = V I∗ = P + jQ V = Vrms = √ V • 2 Cosine Wave RMS • 2 e 2 • Power Factor + e e V e ∗ • Complex Power For an impedance Z: S = I Z = | | ◦ Z • Power in R, L, C e • Tellegen’s Theorem Apparent Power: S = V I used for machine ratings. • Power Factor Correction ◦ | | • Ideal Transformer e e • Average Power: (in Watts) Transformer Applications P = (S) = V I cos φ • Summary ◦ ℜ e e Reactive Power: Q = (S) = V I sin φ (in VARs) ◦ ℑ e e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 11 / 11 Summary
14: Power in AC Circuits 1 • Average Power Complex Power: where . S = V I∗ = P + jQ V = Vrms = √ V • 2 Cosine Wave RMS • 2 e 2 • Power Factor + e e V e ∗ • Complex Power For an impedance Z: S = I Z = | | ◦ Z • Power in R, L, C e • Tellegen’s Theorem Apparent Power: S = V I used for machine ratings. • Power Factor Correction ◦ | | • Ideal Transformer e e • Average Power: (in Watts) Transformer Applications P = (S) = V I cos φ • Summary ◦ ℜ e e Reactive Power: Q = (S) = V I sin φ (in VARs) ◦ ℑ Power engineers always use V and e I eand omit the ~. ◦ e e
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 11 / 11 Summary
14: Power in AC Circuits 1 • Average Power Complex Power: where . S = V I∗ = P + jQ V = Vrms = √ V • 2 Cosine Wave RMS • 2 e 2 • Power Factor + e e V e ∗ • Complex Power For an impedance Z: S = I Z = | | ◦ Z • Power in R, L, C e • Tellegen’s Theorem Apparent Power: S = V I used for machine ratings. • Power Factor Correction ◦ | | • Ideal Transformer e e • Average Power: (in Watts) Transformer Applications P = (S) = V I cos φ • Summary ◦ ℜ e e Reactive Power: Q = (S) = V I sin φ (in VARs) ◦ ℑ Power engineers always use V and e I eand omit the ~. ◦ e e Tellegen: In any circuit Sb = 0 Pb = Qb = 0 • Pb ⇒ Pb Pb
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 11 / 11 Summary
14: Power in AC Circuits 1 • Average Power Complex Power: where . S = V I∗ = P + jQ V = Vrms = √ V • 2 Cosine Wave RMS • 2 e 2 • Power Factor + e e V e ∗ • Complex Power For an impedance Z: S = I Z = | | ◦ Z • Power in R, L, C e • Tellegen’s Theorem Apparent Power: S = V I used for machine ratings. • Power Factor Correction ◦ | | • Ideal Transformer e e • Average Power: (in Watts) Transformer Applications P = (S) = V I cos φ • Summary ◦ ℜ e e Reactive Power: Q = (S) = V I sin φ (in VARs) ◦ ℑ Power engineers always use V and e I eand omit the ~. ◦ e e Tellegen: In any circuit Sb = 0 Pb = Qb = 0 • Pb ⇒ Pb Pb Power Factor Correction: add parallel C to generate extra VARs •
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 11 / 11 Summary
14: Power in AC Circuits 1 • Average Power Complex Power: where . S = V I∗ = P + jQ V = Vrms = √ V • 2 Cosine Wave RMS • 2 e 2 • Power Factor + e e V e ∗ • Complex Power For an impedance Z: S = I Z = | | ◦ Z • Power in R, L, C e • Tellegen’s Theorem Apparent Power: S = V I used for machine ratings. • Power Factor Correction ◦ | | • Ideal Transformer e e • Average Power: (in Watts) Transformer Applications P = (S) = V I cos φ • Summary ◦ ℜ e e Reactive Power: Q = (S) = V I sin φ (in VARs) ◦ ℑ Power engineers always use V and e I eand omit the ~. ◦ e e Tellegen: In any circuit Sb = 0 Pb = Qb = 0 • Pb ⇒ Pb Pb Power Factor Correction: add parallel C to generate extra VARs •
Ideal Transformer: Vi Ni and NiIi = 0 (implies Si = 0) • ∝ P P
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 11 / 11 Summary
14: Power in AC Circuits 1 • Average Power Complex Power: where . S = V I∗ = P + jQ V = Vrms = √ V • 2 Cosine Wave RMS • 2 e 2 • Power Factor + e e V e ∗ • Complex Power For an impedance Z: S = I Z = | | ◦ Z • Power in R, L, C e • Tellegen’s Theorem Apparent Power: S = V I used for machine ratings. • Power Factor Correction ◦ | | • Ideal Transformer e e • Average Power: (in Watts) Transformer Applications P = (S) = V I cos φ • Summary ◦ ℜ e e Reactive Power: Q = (S) = V I sin φ (in VARs) ◦ ℑ Power engineers always use V and e I eand omit the ~. ◦ e e Tellegen: In any circuit Sb = 0 Pb = Qb = 0 • Pb ⇒ Pb Pb Power Factor Correction: add parallel C to generate extra VARs •
Ideal Transformer: Vi Ni and NiIi = 0 (implies Si = 0) • ∝ P P For further details see Hayt Ch 11 or Irwin Ch 9.
E1.1 Analysis of Circuits (2017-10213) AC Power: 14 – 11 / 11