14: Power in AC Circuits. • Average Power. • Cosine Wave RMS. • Power Factor. •
Complex Power. • Power in R, L, C. • Tellegen's Theorem. • 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 • 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 • Ideal Transformer • Transformer Applications • Summary
Intantaneous Power dissipated in R: p(t) =
E1.1 Analysis of Circuits (2017-10213)
v 2 (t) R
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 • Ideal Transformer • Transformer Applications • Summary
Intantaneous Power dissipated in R: p(t) =
E1.1 Analysis of Circuits (2017-10213)
v 2 (t) R
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 • Ideal Transformer • Transformer Applications • Summary
Intantaneous Power dissipated in R: p(t) =
v 2 (t) R
Average Power dissipated in R:
P =
E1.1 Analysis of Circuits (2017-10213)
1 T
RT 0
p(t)dt
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 • Ideal Transformer • Transformer Applications • Summary
Intantaneous Power dissipated in R: p(t) = Average Power dissipated in R:
P =
E1.1 Analysis of Circuits (2017-10213)
1 T
RT 0
p(t)dt =
1 R
×
1 T
RT 0
v 2 (t) R
v 2 (t)dt
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 • Ideal Transformer • Transformer Applications • Summary
Intantaneous Power dissipated in R: p(t) = Average Power dissipated in R:
P =
E1.1 Analysis of Circuits (2017-10213)
1 T
RT 0
p(t)dt =
1 R
×
1 T
RT 0
2
v 2 (t) R
v (t)dt =
hv2 (t)i R
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 • Ideal Transformer • Transformer Applications • Summary
Intantaneous Power dissipated in R: p(t) = Average Power dissipated in R:
R 1 T T 0
1 R
1 T
RT
2
v 2 (t) R
hv2 (t)i
P p(t)dt = × v (t)dt = R 0
2= v (t) is the value of v 2 (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 • Ideal Transformer • Transformer Applications • Summary
Intantaneous Power dissipated in R: p(t) = Average Power dissipated in R:
R 1 T T 0
1 R
1 T
RT
2
v 2 (t) R
hv2 (t)i
P p(t)dt = × v (t)dt = R 0
2= v (t) is the value of v 2 (t) averaged over time
We define the RMS Voltage (Root Mean Square): Vrms
E1.1 Analysis of Circuits (2017-10213)
p , hv 2 (t)i
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 • Ideal Transformer • Transformer Applications • Summary
Intantaneous Power dissipated in R: p(t) = Average Power dissipated in R:
R 1 T T 0
1 R
1 T
RT
2
v 2 (t) R
hv2 (t)i
P p(t)dt = × v (t)dt = R 0
2= v (t) is the value of v 2 (t) averaged over time
We define the RMS Voltage (Root Mean Square): Vrms
hv2 (t)i
p , hv 2 (t)i
(Vrms )2 R
= The average power dissipated in R is P = 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 • Ideal Transformer • Transformer Applications • Summary
Intantaneous Power dissipated in R: p(t) = Average Power dissipated in R:
R 1 T T 0
1 R
1 T
RT
2
v 2 (t) R
hv2 (t)i
P p(t)dt = × v (t)dt = R 0
2= v (t) is the value of v 2 (t) averaged over time
We define the RMS Voltage (Root Mean Square): Vrms
hv2 (t)i
p , hv 2 (t)i
(Vrms )2 R
= The average power dissipated in R is P = 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 • Summary
Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V.
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 • Summary
Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. 2
2
2
Squared Voltage: v (t) = V cos ωt = V
E1.1 Analysis of Circuits (2017-10213)
2
1 2
+
1 2
cos 2ωt
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 • Summary
Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. 2
2
2
Squared Voltage: v (t) = V cos ωt = V
2 Mean Square Voltage: v =
E1.1 Analysis of Circuits (2017-10213)
V2 2
2
1 2
+
1 2
cos 2ωt
since cos 2ωt averages to zero.
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 • Summary
Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. 2
2
2
Squared Voltage: v (t) = V cos ωt = V
2 Mean Square Voltage: v =
E1.1 Analysis of Circuits (2017-10213)
V2 2
2
1 2
+
1 2
cos 2ωt
since cos 2ωt averages to zero.
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 • Summary
Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. 2
2
2
Squared Voltage: v (t) = V cos ωt = V
2
1 2
+
1 2
cos 2ωt
2 V 2 Mean Square Voltage: v = 2 since cos 2ωt averages to zero. p RMS Voltage: Vrms = hv 2 i = √1 V = 3.54 V 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 • Summary
Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. 2
2
2
Squared Voltage: v (t) = V cos ωt = V
2
1 2
+
1 2
cos 2ωt
2 V 2 Mean Square Voltage: v = 2 since cos 2ωt averages to zero. p RMS Voltage: Vrms = hv 2 i = √1 V = 3.54 V 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 • Summary
Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V. 2
2
2
Squared Voltage: v (t) = V cos ωt = V
2
1 2
+
1 2
cos 2ωt
2 V 2 Mean Square Voltage: v = 2 since cos 2ωt averages to zero. p e RMS Voltage: Vrms = hv 2 i = √1 V = 3.54 V = V 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.
√ e = In this lecture course only, a ~ overbar means ÷ 2: thus V E1.1 Analysis of Circuits (2017-10213)
√1 V 2
.
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 • Transformer Applications • Summary
Suppose voltage and current phasors are:
V = |V | ejθV 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 • Transformer Applications • Summary
Suppose voltage and current phasors are:
V = |V | ejθV I = |I| ejθI
E1.1 Analysis of Circuits (2017-10213)
⇔ ⇔
v(t) = |V | cos (ωt + θV ) i(t) = |I| cos (ωt + θI )
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 • Transformer Applications • Summary
Suppose voltage and current phasors are:
V = |V | ejθV I = |I| ejθI
E1.1 Analysis of Circuits (2017-10213)
⇔ ⇔
v(t) = |V | cos (ωt + θV ) i(t) = |I| cos (ωt + θI )
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 • Transformer Applications • Summary
Suppose voltage and current phasors are:
V = |V | ejθV I = |I| ejθI
⇔ ⇔
v(t) = |V | cos (ωt + θV ) 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 • Transformer Applications • Summary
Suppose voltage and current phasors are:
V = |V | ejθV I = |I| ejθI
⇔ ⇔
v(t) = |V | cos (ωt + θV ) 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 ) = |V | |I|
E1.1 Analysis of Circuits (2017-10213)
1 2
cos (2ωt + θV + θI ) +
1 2
cos (θV − θI )
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 • Transformer Applications • Summary
Suppose voltage and current phasors are:
V = |V | ejθV I = |I| ejθI
⇔ ⇔
v(t) = |V | cos (ωt + θV ) 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 2
1 2
= |V | |I| cos (2ωt + θV + θI ) + cos (θV − θI ) = 21 |V | |I| cos (θV − θI ) + 12 |V | |I| cos (2ωt + θV + 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 • Transformer Applications • Summary
Suppose voltage and current phasors are:
V = |V | ejθV I = |I| ejθI
⇔ ⇔
v(t) = |V | cos (ωt + θV ) 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 2
1 2
= |V | |I| cos (2ωt + θV + θI ) + cos (θV − θI ) = 21 |V | |I| cos (θV − θI ) + 12 |V | |I| cos (2ωt + θV + 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 • Transformer Applications • Summary
Suppose voltage and current phasors are:
V = |V | ejθV I = |I| ejθI
⇔ ⇔
v(t) = |V | cos (ωt + θV ) 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 2
1 2
= |V | |I| cos (2ωt + θV + θI ) + cos (θV − θI ) = 21 |V | |I| cos (θV − θI ) + 12 |V | |I| cos (2ωt + θV + θI )
Average power: P = 12 |V | |I| cos (φ)
E1.1 Analysis of Circuits (2017-10213)
where
φ = θV − θI
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 • Transformer Applications • Summary
Suppose voltage and current phasors are:
V = |V | ejθV I = |I| ejθI
⇔ ⇔
v(t) = |V | cos (ωt + θV ) 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 2
1 2
= |V | |I| cos (2ωt + θV + θI ) + cos (θV − θI ) = 21 |V | |I| cos (θV − θI ) + 12 |V | |I| cos (2ωt + θV + θI )
Average power: P = 12 |V | |I| cos (φ)
= Ve Ie cos (φ)
E1.1 Analysis of Circuits (2017-10213)
where
φ = θV − θI
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 • Transformer Applications • Summary
Suppose voltage and current phasors are:
V = |V | ejθV I = |I| ejθI
⇔ ⇔
v(t) = |V | cos (ωt + θV ) 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 2
1 2
= |V | |I| cos (2ωt + θV + θI ) + cos (θV − θI ) = 21 |V | |I| cos (θV − θI ) + 12 |V | |I| cos (2ωt + θV + θI )
Average power: P = 12 |V | |I| cos (φ)
= Ve Ie cos (φ)
E1.1 Analysis of Circuits (2017-10213)
where
φ = θV − θI
cos φ is the power factor
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 • Transformer Applications • Summary
Suppose voltage and current phasors are:
V = |V | ejθV I = |I| ejθI
⇔ ⇔
v(t) = |V | cos (ωt + θV ) 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 2
1 2
= |V | |I| cos (2ωt + θV + θI ) + cos (θV − θI ) = 21 |V | |I| cos (θV − θI ) + 12 |V | |I| cos (2ωt + θV + θI )
Average power: P = 12 |V | |I| cos (φ)
= Ve Ie cos (φ)
where
φ = θV − θI
cos φ is the power factor
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
• 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 5 / 11
Complex 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate. Ve × Ie∗ = Ve ejθV × Ie e−jθI
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 5 / 11
Complex 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
Ve × Ie∗ = Ve ejθV × Ie e−jθI = Ve Ie ej(θV −θI )
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 5 / 11
Complex 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
Ve × Ie∗ = Ve ejθV × Ie e−jθI = Ve Ie ej(θV −θI ) = Ve Ie ejφ
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 5 / 11
Complex 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
Ve × Ie∗ = Ve ejθV × Ie e−jθI = Ve Ie ej(θV −θI ) e e jφ e e = V I e = V I cos φ + j Ve Ie sin φ
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 5 / 11
Complex 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
Ve × Ie∗ = Ve ejθV × Ie e−jθI = Ve Ie ej(θV −θI ) e e jφ e e = V I e = V I cos φ + j Ve Ie sin φ = P + jQ
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 5 / 11
Complex 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
Ve × Ie∗ = Ve ejθV × Ie e−jθI = Ve Ie ej(θV −θI ) e e jφ e e = V I e = V I cos φ + j Ve Ie sin φ = P + jQ
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 5 / 11
Complex 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
Ve × Ie∗ = Ve ejθV × Ie e−jθI = Ve Ie ej(θV −θI ) e e jφ e e = V I e = V I cos φ + j Ve Ie sin φ = P + jQ
Complex Power:
E1.1 Analysis of Circuits (2017-10213)
S , Ve Ie∗ = P + jQ measured in Volt-Amps (VA)
AC Power: 14 – 5 / 11
Complex 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
Ve × Ie∗ = Ve ejθV × Ie e−jθI = Ve Ie ej(θV −θI ) e e jφ e e = V I e = V I cos φ + j Ve Ie sin φ = P + jQ
S , Ve Ie ∗ = P + jQ measured in Volt-Amps (VA) e e Apparent Power: |S| , V I measured in Volt-Amps (VA) Complex Power:
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 5 / 11
Complex 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
Ve × Ie∗ = Ve ejθV × Ie e−jθI = Ve Ie ej(θV −θI ) e e jφ e e = V I e = V I cos φ + j Ve Ie sin φ = P + jQ
S , Ve Ie ∗ = P + jQ measured in Volt-Amps (VA) e e Apparent Power: |S| , V I measured in Volt-Amps (VA) Complex Power:
Average Power:
E1.1 Analysis of Circuits (2017-10213)
P , ℜ (S) measured in Watts (W)
AC Power: 14 – 5 / 11
Complex 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
Ve × Ie∗ = Ve ejθV × Ie e−jθI = Ve Ie ej(θV −θI ) e e jφ e e = V I e = V I cos φ + j Ve Ie sin φ = P + jQ
S , Ve Ie ∗ = P + jQ measured in Volt-Amps (VA) e e Apparent Power: |S| , V I measured in Volt-Amps (VA) Complex Power:
Average Power: P , ℜ (S) measured in Watts (W) Reactive Power: Q , ℑ (S) Measured in Volt-Amps Reactive (VAR)
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 5 / 11
Complex 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
Ve × Ie∗ = Ve ejθV × Ie e−jθI = Ve Ie ej(θV −θI ) e e jφ e e = V I e = V I cos φ + j Ve Ie sin φ = P + jQ
S , Ve Ie ∗ = P + jQ measured in Volt-Amps (VA) e e Apparent Power: |S| , V I measured in Volt-Amps (VA) Complex Power:
Average Power: P , ℜ (S) measured in Watts (W) Reactive Power: Q , ℑ (S) Measuredin Volt-Amps Reactive (VAR)
e − ∠Ie = P Power Factor: cos φ , cos ∠V |S|
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 5 / 11
Complex 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
Ve × Ie∗ = Ve ejθV × Ie e−jθI = Ve Ie ej(θV −θI ) e e jφ e e = V I e = V I cos φ + j Ve Ie sin φ = P + jQ
S , Ve Ie ∗ = P + jQ measured in Volt-Amps (VA) e e Apparent Power: |S| , V I measured in Volt-Amps (VA) Complex Power:
Average Power: P , ℜ (S) measured in Watts (W) Reactive Power: Q , ℑ (S) Measuredin Volt-Amps Reactive (VAR)
e − ∠Ie = P Power Factor: cos φ , cos ∠V |S|
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
• 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
e = √1 |V | ejθV and If V 2
I˜ =
√1 2
|I| ejθI
e × Ie∗ The complex power absorbed by Z is S , V where * means complex conjugate.
Ve × Ie∗ = Ve ejθV × Ie e−jθI = Ve Ie ej(θV −θI ) e e jφ e e = V I e = V I cos φ + j Ve Ie sin φ = P + jQ
S , Ve Ie ∗ = P + jQ measured in Volt-Amps (VA) e e Apparent Power: |S| , V I measured in Volt-Amps (VA) Complex Power:
Average Power: P , ℜ (S) measured in Watts (W) Reactive Power: Q , ℑ (S) Measuredin Volt-Amps Reactive (VAR)
e − ∠Ie = P Power Factor: cos φ , cos ∠V |S|
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 • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
e Ie∗ = P + jQ For any impedance, Z , complex power absorbed: S = V
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 6 / 11
Power in R, L, C 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
e Ie∗ = P + jQ For any impedance, Z , complex power absorbed: S = V 2 2 e |2 V | e = IZ e (b) Ie × Ie∗ = Ie we get S = Ie Z = ∗ Using (a) V Z
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 6 / 11
Power in R, L, C 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
e Ie∗ = P + jQ For any impedance, Z , complex power absorbed: S = V 2 2 e |2 V | e = IZ e (b) Ie × Ie∗ = Ie we get S = Ie Z = ∗ Using (a) V Z
2 2 |Ve | e Resistor: S = I R = R
E1.1 Analysis of Circuits (2017-10213)
φ=0
Absorbs average power, no VARs (Q = 0)
AC Power: 14 – 6 / 11
Power in R, L, C 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
e Ie∗ = P + jQ For any impedance, Z , complex power absorbed: S = V 2 2 e |2 V | e = IZ e (b) Ie × Ie∗ = Ie we get S = Ie Z = ∗ Using (a) V Z
2 2 |Ve | e Resistor: S = I R = R
φ=0
Absorbs average power, no VARs (Q = 0)
2 2 |Ve | e Inductor: S = j I ωL = j ωL
φ = +90◦
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 • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
e Ie∗ = P + jQ For any impedance, Z , complex power absorbed: S = V 2 2 e |2 V | e = IZ e (b) Ie × Ie∗ = Ie we get S = Ie Z = ∗ Using (a) V Z
2 2 |Ve | e Resistor: S = I R = R
φ=0
Absorbs average power, no VARs (Q = 0)
2 2 |Ve | e Inductor: S = j I ωL = j ωL
φ = +90◦
No average power, Absorbs VARs (Q > 0)
|Ie|
2
Capacitor: S = −j ωC
2 = −j Ve ωC
φ = −90◦
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 • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
e Ie∗ = P + jQ For any impedance, Z , complex power absorbed: S = V 2 2 e |2 V | e = IZ e (b) Ie × Ie∗ = Ie we get S = Ie Z = ∗ Using (a) V Z
2 2 |Ve | e Resistor: S = I R = R
φ=0
Absorbs average power, no VARs (Q = 0)
2 2 |Ve | e Inductor: S = j I ωL = j ωL
φ = +90◦
No average power, Absorbs VARs (Q > 0)
|Ie|
2
Capacitor: S = −j ωC
2 = −j Ve ωC
φ = −90◦
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 • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
e Ie∗ = P + jQ For any impedance, Z , complex power absorbed: S = V 2 2 e |2 V | e = IZ e (b) Ie × Ie∗ = Ie we get S = Ie Z = ∗ Using (a) V Z
2 2 |Ve | e Resistor: S = I R = R
φ=0
Absorbs average power, no VARs (Q = 0)
2 2 |Ve | e Inductor: S = j I ωL = j ωL
φ = +90◦
No average power, Absorbs VARs (Q > 0)
|Ie|
2
Capacitor: S = −j ωC
2 = −j Ve ωC
φ = −90◦
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
• 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
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s components sums to zero.
xn = voltage at node n Vb , Ib = voltage/current in branch b
E1.1 Analysis of Circuits (2017-10213)
(obeying passive sign convention)
AC Power: 14 – 7 / 11
Tellegen’s Theorem 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
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s components sums to zero.
xn = voltage at node n Vb , Ib = voltage/current in branch b (obeying passive sign convention)
abn
−1 if Vb starts from node n , +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
• 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
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s components sums to zero.
xn = voltage at node n Vb , Ib = voltage/current in branch b (obeying passive sign convention)
abn
−1 if Vb starts from node n , +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
• 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
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s components sums to zero.
xn = voltage at node n Vb , Ib = voltage/current in branch b (obeying passive sign convention)
abn
−1 if Vb starts from node n , +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 =
E1.1 Analysis of Circuits (2017-10213)
P
n
abn xn (e.g. V4 = x3 − x2 )
AC Power: 14 – 7 / 11
Tellegen’s Theorem 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
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s components sums to zero.
xn = voltage at node n Vb , Ib = voltage/current in branch b (obeying passive sign convention)
abn
−1 if Vb starts from node n , +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 =
P
abn xn (e.g. V4 = x3 − x2 ) P P ∗ KCL @ node n: b abn Ib = 0 ⇒ b abn Ib = 0
E1.1 Analysis of Circuits (2017-10213)
n
AC Power: 14 – 7 / 11
Tellegen’s Theorem 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
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s components sums to zero.
xn = voltage at node n Vb , Ib = voltage/current in branch b (obeying passive sign convention)
abn
−1 if Vb starts from node n , +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 =
P
abn xn (e.g. V4 = x3 − x2 ) P P ∗ KCL @ node n: b abn Ib = 0 ⇒ b abn Ib = 0 P P P ∗ ∗ Tellegen: b V b Ib = b n abn xn Ib
E1.1 Analysis of Circuits (2017-10213)
n
AC Power: 14 – 7 / 11
Tellegen’s Theorem 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
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s components sums to zero.
xn = voltage at node n Vb , Ib = voltage/current in branch b (obeying passive sign convention)
abn
−1 if Vb starts from node n , +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 =
P
abn xn (e.g. V4 = x3 − x2 ) P P ∗ KCL @ node n: b abn Ib = 0 ⇒ b abn Ib = 0 P P P ∗ ∗ Tellegen: b V b Ib = b n abn xn Ib P P = n b abn Ib∗ xn E1.1 Analysis of Circuits (2017-10213)
n
AC Power: 14 – 7 / 11
Tellegen’s Theorem 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
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s components sums to zero.
xn = voltage at node n Vb , Ib = voltage/current in branch b (obeying passive sign convention)
abn
−1 if Vb starts from node n , +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 =
P
abn xn (e.g. V4 = x3 − x2 ) P P ∗ KCL @ node n: b abn Ib = 0 ⇒ b abn Ib = 0 P P P ∗ ∗ Tellegen: b V b Ib = b n abn xn Ib P P P P ∗ = n b abn Ib xn = n xn b abn Ib∗ E1.1 Analysis of Circuits (2017-10213)
n
AC Power: 14 – 7 / 11
Tellegen’s Theorem 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
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s components sums to zero.
xn = voltage at node n Vb , Ib = voltage/current in branch b (obeying passive sign convention)
abn
−1 if Vb starts from node n , +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 =
P
abn xn (e.g. V4 = x3 − x2 ) P P ∗ KCL @ node n: b abn Ib = 0 ⇒ b abn Ib = 0 P P P ∗ ∗ Tellegen: b V b Ib = b n abn xn Ib P P P P P ∗ ∗ = n b abn Ib xn = n xn b abn Ib = n xn × 0 E1.1 Analysis of Circuits (2017-10213)
n
AC Power: 14 – 7 / 11
Tellegen’s Theorem 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
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s components sums to zero.
xn = voltage at node n Vb , Ib = voltage/current in branch b (obeying passive sign convention)
abn
−1 if Vb starts from node n , +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 =
P
abn xn (e.g. V4 = x3 − x2 ) P P ∗ KCL @ node n: b abn Ib = 0 ⇒ b abn Ib = 0 P P P ∗ ∗ Tellegen: b V b Ib = b n abn xn Ib P P P P P ∗ ∗ = n b abn Ib xn = n xn b abn Ib = n xn × 0 P P P Note: and also b Sb = 0 ⇒ b Pb = 0 b Qb = 0. E1.1 Analysis of Circuits (2017-10213)
n
AC Power: 14 – 7 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S|
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S| Add parallel capacitor of 300 µF:
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S| Add parallel capacitor of 300 µF:
ZC =
E1.1 Analysis of Circuits (2017-10213)
1 jωC
= −10.6j Ω
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S| Add parallel capacitor of 300 µF:
ZC =
E1.1 Analysis of Circuits (2017-10213)
1 jωC
= −10.6j Ω ⇒ IeC = 21.7j A
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S| Add parallel capacitor of 300 µF:
1 = −10.6j Ω ⇒ IeC = 21.7j A ZC = jωC Ie = 46 − j11.2 A = 47∠ − 14◦ A
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S| Add parallel capacitor of 300 µF:
1 = −10.6j Ω ⇒ IeC = 21.7j A ZC = jωC Ie = 46 − j11.2 A = 47∠ − 14◦ A
SC = Ve IeC∗ = −j5 kVA
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S| Add parallel capacitor of 300 µF:
1 = −10.6j Ω ⇒ IeC = 21.7j A ZC = jωC Ie = 46 − j11.2 A = 47∠ − 14◦ A
SC = Ve IeC∗ = −j5 kVA S = Ve Ie∗ = 10.6 + j2.6 kVA
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S| Add parallel capacitor of 300 µF:
1 = −10.6j Ω ⇒ IeC = 21.7j A ZC = jωC Ie = 46 − j11.2 A = 47∠ − 14◦ A
SC = Ve IeC∗ = −j5 kVA S = Ve Ie∗ = 10.6 + j2.6 kVA = 10.9∠14◦ kVA
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S| Add parallel capacitor of 300 µF:
1 = −10.6j Ω ⇒ IeC = 21.7j A ZC = jωC Ie = 46 − j11.2 A = 47∠ − 14◦ A
SC = Ve IeC∗ = −j5 kVA S = Ve Ie∗ = 10.6 + j2.6 kVA = 10.9∠14◦ kVA P cos φ = |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 • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S| Add parallel capacitor of 300 µF:
1 = −10.6j Ω ⇒ IeC = 21.7j A ZC = jωC Ie = 46 − j11.2 A = 47∠ − 14◦ A
SC = Ve IeC∗ = −j5 kVA S = Ve Ie∗ = 10.6 + j2.6 kVA = 10.9∠14◦ kVA P cos φ = |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 • Cosine Wave RMS • Power Factor + • Complex Power • Power in R, L, C • Tellegen’s Theorem • Power Factor Correction • Ideal Transformer • Transformer Applications • Summary
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S| Add parallel capacitor of 300 µF:
1 = −10.6j Ω ⇒ IeC = 21.7j A ZC = jωC Ie = 46 − j11.2 A = 47∠ − 14◦ A
SC = Ve IeC∗ = −j5 kVA S = Ve Ie∗ = 10.6 + j2.6 kVA = 10.9∠14◦ kVA P cos φ = |S| = cos 14◦ = 0.97 power to motor, P , is 10.6 kW in both cases. Average e I , reduced from 56.5 ց 47 A (−16%) ⇒ lower losses. E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S| Add parallel capacitor of 300 µF:
1 = −10.6j Ω ⇒ IeC = 21.7j A ZC = jωC Ie = 46 − j11.2 A = 47∠ − 14◦ A
SC = Ve IeC∗ = −j5 kVA S = Ve Ie∗ = 10.6 + j2.6 kVA = 10.9∠14◦ kVA P cos φ = |S| = cos 14◦ = 0.97 power to motor, P , is 10.6 kW in both cases. Average e I , reduced from 56.5 ց 47 A (−16%) ⇒ lower losses. Effect of C : VARs = 7.6 ց 2.6 kVAR E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Power Factor Correction 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
Ve = 230. Motor modelled as 5||7j Ω. e e Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦ S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA P = cos 36◦ = 0.81 cos φ = |S| Add parallel capacitor of 300 µF:
1 = −10.6j Ω ⇒ IeC = 21.7j A ZC = jωC Ie = 46 − j11.2 A = 47∠ − 14◦ A
SC = Ve IeC∗ = −j5 kVA S = Ve Ie∗ = 10.6 + j2.6 kVA = 10.9∠14◦ kVA P cos φ = |S| = cos 14◦ = 0.97 power to motor, P , is 10.6 kW in both cases. Average e I , reduced from 56.5 ց 47 A (−16%) ⇒ lower losses. Effect of C : VARs = 7.6 ց 2.6 kVAR , power factor = 0.81 ր 0.97. E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 8 / 11
Ideal Transformer 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
A transformer has ≥ 2 windings on the same magnetic core.
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 9 / 11
Ideal Transformer 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
A transformer has ≥ 2 windings on the same magnetic core. Ampère’s law:
E1.1 Analysis of Circuits (2017-10213)
P
Nr Ir =
lΦ µA ;
Vr dΦ = Faraday’s law: N dt . r
AC Power: 14 – 9 / 11
Ideal Transformer 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
A transformer has ≥ 2 windings on the same magnetic core.
P
Vr dΦ = Faraday’s law: N dt . r N1 : N2 + N3 shows the turns ratio between the windings. The • indicates the voltage polarity of each winding.
Ampère’s law:
E1.1 Analysis of Circuits (2017-10213)
Nr Ir =
lΦ µA ;
AC Power: 14 – 9 / 11
Ideal Transformer 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
A transformer has ≥ 2 windings on the same magnetic core.
P
Vr dΦ = Faraday’s law: N dt . r N1 : N2 + N3 shows the turns ratio between the windings. The • indicates the voltage polarity of each winding.
Ampère’s law:
Nr Ir =
lΦ µA ;
V1 V2 V3 Since Φ is the same for all windings, N = = N2 N3 . 1
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 9 / 11
Ideal Transformer 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
A transformer has ≥ 2 windings on the same magnetic core.
P
Vr dΦ = Faraday’s law: N dt . r N1 : N2 + N3 shows the turns ratio between the windings. The • indicates the voltage polarity of each winding.
Ampère’s law:
Nr Ir =
lΦ µA ;
V1 V2 V3 Since Φ is the same for all windings, N = = N2 N3 . 1
Assume µ → ∞ ⇒ N1 I1 + N2 I2 + N3 I3 = 0
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 9 / 11
Ideal Transformer 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
A transformer has ≥ 2 windings on the same magnetic core.
P
Vr dΦ = Faraday’s law: N dt . r N1 : N2 + N3 shows the turns ratio between the windings. The • indicates the voltage polarity of each winding.
Ampère’s law:
Nr Ir =
lΦ µA ;
V1 V2 V3 Since Φ is the same for all windings, N = = N2 N3 . 1
Assume µ → ∞ ⇒ N1 I1 + N2 I2 + N3 I3 = 0 These two equations allow you to solve circuits and also P imply that Si = 0.
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 9 / 11
Ideal Transformer 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
A transformer has ≥ 2 windings on the same magnetic core.
P
Vr dΦ = Faraday’s law: N dt . r N1 : N2 + N3 shows the turns ratio between the windings. The • indicates the voltage polarity of each winding.
Ampère’s law:
Nr Ir =
lΦ µA ;
V1 V2 V3 Since Φ is the same for all windings, N = = N2 N3 . 1
Assume µ → ∞ ⇒ N1 I1 + N2 I2 + N3 I3 = 0 These two equations allow you to solve circuits and also P imply that Si = 0. Special Case:
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 9 / 11
Ideal Transformer 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
A transformer has ≥ 2 windings on the same magnetic core.
P
Vr dΦ = Faraday’s law: N dt . r N1 : N2 + N3 shows the turns ratio between the windings. The • indicates the voltage polarity of each winding.
Ampère’s law:
Nr Ir =
lΦ µA ;
V1 V2 V3 Since Φ is the same for all windings, N = = N2 N3 . 1
Assume µ → ∞ ⇒ N1 I1 + N2 I2 + N3 I3 = 0 These two equations allow you to solve circuits and also P imply that Si = 0. Special Case: For a 2-winding transformer this simplifies to N2 N1 V2 = N V1 and IL = −I2 = N I1 1
E1.1 Analysis of Circuits (2017-10213)
2
AC Power: 14 – 9 / 11
Ideal Transformer 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
A transformer has ≥ 2 windings on the same magnetic core.
P
Vr dΦ = Faraday’s law: N dt . r N1 : N2 + N3 shows the turns ratio between the windings. The • indicates the voltage polarity of each winding.
Ampère’s law:
Nr Ir =
lΦ µA ;
V1 V2 V3 Since Φ is the same for all windings, N = = N2 N3 . 1
Assume µ → ∞ ⇒ N1 I1 + N2 I2 + N3 I3 = 0 These two equations allow you to solve circuits and also P imply that Si = 0. Special Case: For a 2-winding transformer this simplifies to N2 N1 V2 = N V1 and IL = −I2 = N I1 1
Hence VI 1 = 1
E1.1 Analysis of Circuits (2017-10213)
2
N1 N2
2
V2 IL
=
N1 N2
2
Z
AC Power: 14 – 9 / 11
Ideal Transformer 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
A transformer has ≥ 2 windings on the same magnetic core.
P
Vr dΦ = Faraday’s law: N dt . r N1 : N2 + N3 shows the turns ratio between the windings. The • indicates the voltage polarity of each winding.
Ampère’s law:
Nr Ir =
lΦ µA ;
V1 V2 V3 Since Φ is the same for all windings, N = = N2 N3 . 1
Assume µ → ∞ ⇒ N1 I1 + N2 I2 + N3 I3 = 0 These two equations allow you to solve circuits and also P imply that Si = 0. Special Case: For a 2-winding transformer this simplifies to N2 N1 V2 = N V1 and IL = −I2 = N I1 1
Hence VI 1 = 1
2
N1 N2
2
V2 IL
=
N1 N2
2
Z
Equivalent to a reflected impedance of E1.1 Analysis of Circuits (2017-10213)
N1 N2
2
Z 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
• 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
Power Transmission Suppose a power transmission cable has 1 Ω resistance. 100 kVA@ 1 kV = 100 A ⇒ Ie2 R = 10 kW losses.
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 10 / 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
Power Transmission Suppose a power transmission cable has 1 Ω resistance. 100 kVA@ 1 kV = 100 A ⇒ Ie2 R = 10 kW losses. 100 kVA@ 100 kV = 1 A ⇒ Ie2 R = 1 W losses.
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 10 / 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
Power Transmission Suppose a power transmission cable has 1 Ω resistance. 100 kVA@ 1 kV = 100 A ⇒ Ie2 R = 10 kW losses. 100 kVA@ 100 kV = 1 A ⇒ Ie2 R = 1 W losses.
Voltage Conversion
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
• 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
Power Transmission Suppose a power transmission cable has 1 Ω resistance. 100 kVA@ 1 kV = 100 A ⇒ Ie2 R = 10 kW losses. 100 kVA@ 100 kV = 1 A ⇒ Ie2 R = 1 W losses.
Voltage Conversion
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
• 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
Power Transmission Suppose a power transmission cable has 1 Ω resistance. 100 kVA@ 1 kV = 100 A ⇒ Ie2 R = 10 kW losses. 100 kVA@ 100 kV = 1 A ⇒ Ie2 R = 1 W losses.
Voltage Conversion
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
• 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
• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms =
E1.1 Analysis of Circuits (2017-10213)
√1 V 2
.
AC Power: 14 – 11 / 11
Summary 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
• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = 2 2 |Ve | e ◦ For an impedance Z : S = I Z = Z ∗
E1.1 Analysis of Circuits (2017-10213)
√1 V 2
.
AC Power: 14 – 11 / 11
Summary 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
• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V . 2 2 |Ve | e ◦ For an impedance Z : S = I Z = Z ∗ ◦ Apparent Power: |S| = Ve Ie used for machine ratings.
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 11 / 11
Summary 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
• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V . 2 2 |Ve | e ◦ For an impedance Z : S = I Z = Z ∗ ◦ Apparent Power: |S| = Ve Ie used for machine ratings. ◦ Average Power: P = ℜ (S) = Ve Ie cos φ (in Watts) ◦ Reactive Power: Q = ℑ (S) = Ve Ie sin φ (in VARs)
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 11 / 11
Summary 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
• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V . 2 2 |Ve | e ◦ For an impedance Z : S = I Z = Z ∗ ◦ Apparent Power: |S| = Ve Ie used for machine ratings. ◦ Average Power: P = ℜ (S) = Ve Ie cos φ (in Watts) ◦ Reactive Power: Q = ℑ (S) = Ve Ie sin φ (in VARs) ◦ Power engineers always use Ve and Ie and omit the ~.
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 11 / 11
Summary 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
• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V . 2 2 |Ve | e ◦ For an impedance Z : S = I Z = Z ∗ ◦ Apparent Power: |S| = Ve Ie used for machine ratings. ◦ Average Power: P = ℜ (S) = Ve Ie cos φ (in Watts) ◦ Reactive Power: Q = ℑ (S) = Ve Ie sin φ (in VARs) ◦ Power engineers always use Ve and Ie and omit the ~. • Tellegen: In any circuit
E1.1 Analysis of Circuits (2017-10213)
P
b
Sb = 0 ⇒
P
b
Pb =
P
b
Qb = 0
AC Power: 14 – 11 / 11
Summary 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
• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V . 2 2 |Ve | e ◦ For an impedance Z : S = I Z = Z ∗ ◦ Apparent Power: |S| = Ve Ie used for machine ratings. ◦ Average Power: P = ℜ (S) = Ve Ie cos φ (in Watts) ◦ Reactive Power: Q = ℑ (S) = Ve Ie sin φ (in VARs) ◦ Power engineers always use Ve and Ie and omit the ~. • Tellegen: In any circuit
P
b
Sb = 0 ⇒
P
b
Pb =
P
b
Qb = 0
• 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
• 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
• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V . 2 2 |Ve | e ◦ For an impedance Z : S = I Z = Z ∗ ◦ Apparent Power: |S| = Ve Ie used for machine ratings. ◦ Average Power: P = ℜ (S) = Ve Ie cos φ (in Watts) ◦ Reactive Power: Q = ℑ (S) = Ve Ie sin φ (in VARs) ◦ Power engineers always use Ve and Ie and omit the ~. • Tellegen: In any circuit
P
b
Sb = 0 ⇒
P
b
Pb =
P
b
Qb = 0
• Power Factor Correction: add parallel C to generate extra VARs • Ideal Transformer: Vi ∝ Ni and
E1.1 Analysis of Circuits (2017-10213)
P
Ni Ii = 0 (implies
P
Si = 0)
AC Power: 14 – 11 / 11
Summary 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
• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V . 2 2 |Ve | e ◦ For an impedance Z : S = I Z = Z ∗ ◦ Apparent Power: |S| = Ve Ie used for machine ratings. ◦ Average Power: P = ℜ (S) = Ve Ie cos φ (in Watts) ◦ Reactive Power: Q = ℑ (S) = Ve Ie sin φ (in VARs) ◦ Power engineers always use Ve and Ie and omit the ~. • Tellegen: In any circuit
P
b
Sb = 0 ⇒
P
b
Pb =
P
b
Qb = 0
• Power Factor Correction: add parallel C to generate extra VARs • Ideal Transformer: Vi ∝ Ni and
P
Ni Ii = 0 (implies
P
Si = 0)
For further details see Hayt Ch 11 or Irwin Ch 9.
E1.1 Analysis of Circuits (2017-10213)
AC Power: 14 – 11 / 11