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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
√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 (2015-7265)
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 (2015-7265)
⇔
⇔
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 (2015-7265)
⇔
⇔
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
(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
E1.1 Analysis of Circuits (2015-7265)


−1 if Vb starts from node n
, +1 if Vb ends at node n


0
else
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
√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 (2015-7265)
√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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 (2015-7265)
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 et al. Chapter 11.
E1.1 Analysis of Circuits (2015-7265)
AC Power: 14 – 11 / 11