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

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 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

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 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 v 2 (t)
• Transformer Applications dissipated in R: p(t) = R
• Summary

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 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 v 2 (t)
• Transformer Applications dissipated in R: p(t) = R
• Summary

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 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 v 2 (t)
• Transformer Applications dissipated in R: p(t) = R
• Summary

Average Power dissipated in R:

1
RT
P = T 0
p(t)dt

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 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 v 2 (t)
• Transformer Applications dissipated in R: p(t) = R
• Summary

Average Power dissipated in R:

1
RT 1 1
RT
P = T 0
p(t)dt = R × T 0
v 2 (t)dt

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 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 v 2 (t)
• Transformer Applications dissipated in R: p(t) = R
• Summary

Average Power dissipated in R:

1
RT 1 1
RT 2 hv2 (t)i
P = T 0
p(t)dt = R × T 0
v (t)dt = R

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 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 v 2 (t)
• Transformer Applications Intantaneous Power dissipated in R: p(t) = R
• Summary

Average Power dissipated in R:

R
1 T 1 1
RT 2 hv2 (t)i

P 2= T 0 p(t)dt = ×R T 0
v (t)dt = R
v (t) is the value of v 2 (t) averaged over time

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 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 v 2 (t)
• Transformer Applications Intantaneous Power dissipated in R: p(t) = R
• Summary

Average Power dissipated in R:

R
1 T 1 1
RT 2 hv2 (t)i

P 2= T 0 p(t)dt = ×R T 0
v (t)dt = R
v (t) is the value of v 2 (t) averaged over time
p
We define the RMS Voltage (Root Mean Square): Vrms , hv 2 (t)i

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 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 v 2 (t)
• Transformer Applications Intantaneous Power dissipated in R: p(t) = R
• Summary

Average Power dissipated in R:

R
1 T 1 1
RT 2 hv2 (t)i

P 2= T 0 p(t)dt = ×R T 0
v (t)dt = R
v (t) is the value of v 2 (t) averaged over time
p
We define the RMS Voltage (Root Mean Square): Vrms , hv 2 (t)i
hv2 (t)i (Vrms )2
The average power dissipated in R is P = R = R
Vrms is the DC voltage that would cause R to dissipate the same power.

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 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 v 2 (t)
• Transformer Applications Intantaneous Power dissipated in R: p(t) = R
• Summary

Average Power dissipated in R:

R
1 T 1 1
RT 2 hv2 (t)i

P 2= T 0 p(t)dt = × R T 0
v (t)dt = R
v (t) is the value of v 2 (t) averaged over time
p
We define the RMS Voltage (Root Mean Square): Vrms , hv 2 (t)i
hv2 (t)i (Vrms )2
The average power dissipated in R is P = R = 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.

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 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 : v(t) = 5 cos ωt. Amplitude is V = 5 V.
• Summary

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 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 : v(t) = 5 cos ωt. Amplitude is V = 5 V.
• Summary
2 2 2 2 1 1


Squared Voltage: v (t) = V cos ωt = V 2 + 2 cos 2ωt

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 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 : v(t) = 5 cos ωt. Amplitude is V = 5 V.
• Summary
2 2 2 2 1 1


Squared Voltage: v (t) = V cos ωt = V 2 + 2 cos 2ωt

2 V2
Mean Square Voltage: v = 2 since cos 2ωt averages to zero.

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 Cosine Wave RMS

14: Power in AC Circuits

• Average Power
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V.
• Summary
2 2 2 2 1 1


Squared Voltage: v (t) = V cos ωt = V 2 + 2 cos 2ωt

2 V2
Mean Square Voltage: v = 2 since cos 2ωt averages to zero.

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 Cosine Wave RMS

14: Power in AC Circuits

• Average Power
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V.
• Summary
2 2 2 2 1 1


Squared Voltage: v (t) = V cos ωt = V 2 + 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

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 Cosine Wave RMS

14: Power in AC Circuits

• Average Power
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V.
• Summary
2 2 2 2 1 1


Squared Voltage: v (t) = V cos ωt = V 2 + 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.

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 Cosine Wave RMS

14: Power in AC Circuits

• Average Power
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications Cosine Wave: v(t) = 5 cos ωt. Amplitude is V = 5 V.
• Summary
2 2 2 2 1 1


Squared Voltage: v (t) = V cos ωt = V 2 + 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 √1 V .
2

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 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 V = |V | ejθV
I = |I| ejθI

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 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 V = |V | ejθV ⇔ v(t) = |V | cos (ωt + θV )
I = |I| ejθI ⇔ i(t) = |I| cos (ωt + θI )

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 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 V = |V | ejθV ⇔ v(t) = |V | cos (ωt + θV )
I = |I| ejθI ⇔ i(t) = |I| cos (ωt + θI )

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 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 V = |V | ejθV ⇔ v(t) = |V | cos (ωt + θV )
I = |I| ejθI ⇔ 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 )

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 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 V = |V | ejθV ⇔ v(t) = |V | cos (ωt + θV )
I = |I| ejθI ⇔ i(t) = |I| cos (ωt + θI )
Power dissipated in load Z is
p(t) = v(t)i(t) = |V | |I| cos (ωt + θV ) cos (ωt + θI )
1 1


= |V | |I| 2 cos (2ωt + θV + θI ) + 2 cos (θV − θI )

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 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 V = |V | ejθV ⇔ v(t) = |V | cos (ωt + θV )
I = |I| ejθI ⇔ i(t) = |I| cos (ωt + θI )
Power dissipated in load Z is
p(t) = v(t)i(t) = |V | |I| cos (ωt + θV ) cos (ωt + θI )
1 1


= |V | |I| cos (2ωt + θV + θI ) + cos (θV − θI )
2 2
= 12 |V | |I| cos (θV − θI ) + 12 |V | |I| cos (2ωt + θV + θI )

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 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 V = |V | ejθV ⇔ v(t) = |V | cos (ωt + θV )
I = |I| ejθI ⇔ i(t) = |I| cos (ωt + θI )
Power dissipated in load Z is
p(t) = v(t)i(t) = |V | |I| cos (ωt + θV ) cos (ωt + θI )
1 1


= |V | |I| cos (2ωt + θV + θI ) + cos (θV − θI )
2 2
= 12 |V | |I| cos (θV − θI ) + 12 |V | |I| cos (2ωt + θV + θI )

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 Power Factor

14: Power in AC Circuits

• Average Power
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer Suppose voltage and current phasors are:
• Transformer Applications
• Summary V = |V | ejθV ⇔ v(t) = |V | cos (ωt + θV )
I = |I| ejθI ⇔ i(t) = |I| cos (ωt + θI )
Power dissipated in load Z is
p(t) = v(t)i(t) = |V | |I| cos (ωt + θV ) cos (ωt + θI )
1 1


= |V | |I| cos (2ωt + θV + θI ) + cos (θV − θI )
2 2
= 21 |V | |I| cos (θV − θI ) + 12 |V | |I| cos (2ωt + θV + θI )
Average power: P = 12 |V | |I| cos (φ) where φ = θV − θI

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 Power Factor

14: Power in AC Circuits

• Average Power
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer Suppose voltage and current phasors are:
• Transformer Applications
• Summary V = |V | ejθV ⇔ v(t) = |V | cos (ωt + θV )
I = |I| ejθI ⇔ i(t) = |I| cos (ωt + θI )
Power dissipated in load Z is
p(t) = v(t)i(t) = |V | |I| cos (ωt + θV ) cos (ωt + θI )
1 1


= |V | |I| cos (2ωt + θV + θI ) + cos (θV − θI )
2 2
= 21 |V | |I| cos (θV − θI ) + 12 |V | |I| cos (2ωt + θV + θI )
Average power: P = 12 |V | |I| cos (φ) where φ = θV − θI
  
  
= Ve  Ie cos (φ)

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 Power Factor

14: Power in AC Circuits

• Average Power
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer Suppose voltage and current phasors are:
• Transformer Applications
• Summary V = |V | ejθV ⇔ v(t) = |V | cos (ωt + θV )
I = |I| ejθI ⇔ i(t) = |I| cos (ωt + θI )
Power dissipated in load Z is
p(t) = v(t)i(t) = |V | |I| cos (ωt + θV ) cos (ωt + θI )
1 1


= |V | |I| cos (2ωt + θV + θI ) + cos (θV − θI )
2 2
= 21 |V | |I| cos (θV − θI ) + 12 |V | |I| cos (2ωt + θV + θI )
Average power: P = 12 |V | |I| cos (φ) where φ = θV − θI
  
  
= Ve  Ie cos (φ) cos φ is the power factor

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 Power Factor

14: Power in AC Circuits

• Average Power
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer Suppose voltage and current phasors are:
• Transformer Applications
• Summary V = |V | ejθV ⇔ v(t) = |V | cos (ωt + θV )
I = |I| ejθI ⇔ i(t) = |I| cos (ωt + θI )
Power dissipated in load Z is
p(t) = v(t)i(t) = |V | |I| cos (ωt + θV ) cos (ωt + θI )
1 1


= |V | |I| cos (2ωt + θV + θI ) + cos (θV − θI )
2 2
= 21 |V | |I| cos (θV − θI ) + 12 |V | |I| cos (2ωt + θV + θI )
Average power: P = 12 |V | |I| cos (φ) where φ = θV − θI
  
  
= Ve  Ie cos (φ) 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)
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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
   
• Power Factor Correction
   
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI
• Transformer Applications
• Summary

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
      
• Power Factor Correction
      
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI = Ve  Ie ej(θV −θI )
• Transformer Applications
• Summary

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
      
• Power Factor Correction
      
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI = Ve  Ie ej(θV −θI )
• Transformer Applications
  
• Summary
  
= Ve  Ie ejφ

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
      
• Power Factor Correction
      
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI = Ve  Ie ej(θV −θI )
• Transformer Applications
        
• Summary
 e   e jφ  e   e   
= V  I  e = V  I  cos φ + j Ve  Ie sin φ

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
      
• Power Factor Correction
      
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI = Ve  Ie ej(θV −θI )
• Transformer Applications
        
• Summary
 e   e jφ  e   e   
= V  I  e = V  I  cos φ + j Ve  Ie sin φ

= P + jQ

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
      
• Power Factor Correction
      
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI = Ve  Ie ej(θV −θI )
• Transformer Applications
        
• Summary
 e   e jφ  e   e   
= V  I  e = V  I  cos φ + j Ve  Ie sin φ

= P + jQ

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
      
• Power Factor Correction
      
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI = Ve  Ie ej(θV −θI )
• Transformer Applications
        
• Summary
 e   e jφ  e   e   
= V  I  e = V  I  cos φ + j Ve  Ie sin φ

= P + jQ
Complex Power: S , Ve Ie∗ = P + jQ measured in Volt-Amps (VA)

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
      
• Power Factor Correction
      
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI = Ve  Ie ej(θV −θI )
• Transformer Applications
        
• Summary
 e   e jφ  e   e   
= V  I  e = V  I  cos φ + j Ve  Ie sin φ

= P + jQ
Complex Power: S , Ve Ie∗ = P + jQ measured in Volt-Amps (VA)
 e   e
Apparent Power: |S| , V  I  measured in Volt-Amps (VA)

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
      
• Power Factor Correction
      
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI = Ve  Ie ej(θV −θI )
• Transformer Applications
        
• Summary
 e   e jφ  e   e   
= V  I  e = V  I  cos φ + j Ve  Ie sin φ

= P + jQ
Complex Power: S , Ve Ie∗ = P + jQ measured in Volt-Amps (VA)
 e   e
Apparent Power: |S| , V  I  measured in Volt-Amps (VA)
Average Power: P , ℜ (S) measured in Watts (W)

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
      
• Power Factor Correction
      
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI = Ve  Ie ej(θV −θI )
• Transformer Applications
        
• Summary
 e   e jφ  e   e   
= V  I  e = V  I  cos φ + j Ve  Ie sin φ

= P + jQ
Complex Power: S , Ve Ie∗ = P + jQ measured in Volt-Amps (VA)
 e   e
Apparent Power: |S| , V  I  measured in Volt-Amps (VA)
Average Power: P , ℜ (S) measured in Watts (W)
Reactive Power: Q , ℑ (S) Measured in Volt-Amps Reactive (VAR)

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
      
• Power Factor Correction
      
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI = Ve  Ie ej(θV −θI )
• Transformer Applications
        
• Summary
 e   e jφ  e   e   
= V  I  e = V  I  cos φ + j Ve  Ie sin φ

= P + jQ
Complex Power: S , Ve Ie∗ = P + jQ measured in Volt-Amps (VA)
 e   e
Apparent Power: |S| , V  I  measured in Volt-Amps (VA)
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|

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
      
• Power Factor Correction
      
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI = Ve  Ie ej(θV −θI )
• Transformer Applications
        
• Summary
 e   e jφ  e   e   
= V  I  e = V  I  cos φ + j Ve  Ie sin φ

= P + jQ
Complex Power: S , Ve Ie∗ = P + jQ measured in Volt-Amps (VA)
 e   e
Apparent Power: |S| , V  I  measured in Volt-Amps (VA)
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.

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 Complex Power

e = √1 |V | ejθV and I˜ =
14: Power in AC Circuits
• Average Power If V √1 |I| ejθI
2 2
• Cosine Wave RMS
• Power Factor
• Complex Power
e × Ie∗
The complex power absorbed by Z is S , V
• Power in R, L, C
• Tellegen’s Theorem
where * means complex conjugate.
      
• Power Factor Correction
      
• Ideal Transformer Ve × Ie∗ = Ve  ejθV × Ie e−jθI = Ve  Ie ej(θV −θI )
• Transformer Applications
        
• Summary
 e   e jφ  e   e   
= V  I  e = V  I  cos φ + j Ve  Ie sin φ

= P + jQ
Complex Power: S , Ve Ie∗ = P + jQ measured in Volt-Amps (VA)
 e   e
Apparent Power: |S| , V  I  measured in Volt-Amps (VA)
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.
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 Power in R, L, C

14: Power in AC Circuits

• Average Power
e Ie∗ = P + jQ
For any impedance, Z , complex power absorbed: S = V
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Power in R, L, C

14: Power in AC Circuits

• Average Power
e Ie∗ = P + jQ
For any impedance, Z , complex power absorbed: S = V
 2  2 e |2
• Cosine Wave RMS
e = IZ     |
e (b) Ie × Ie∗ = Ie we get S = Ie Z = ∗V
• Power Factor Using (a) V Z
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Power in R, L, C

14: Power in AC Circuits

• Average Power
e Ie∗ = P + jQ
For any impedance, Z , complex power absorbed: S = V
 2  2 e |2
• Cosine Wave RMS
e = IZ     |
e (b) Ie × Ie∗ = Ie we get S = Ie Z = ∗V
• Power Factor Using (a) V Z
• Complex Power
• Power in R, L, C
 2 2
• Tellegen’s Theorem
 e |Ve |
• Power Factor Correction Resistor: S = I  R = R φ=0
• Ideal Transformer
• Transformer Applications
• Summary Absorbs average power, no VARs (Q = 0)

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 Power in R, L, C

14: Power in AC Circuits

• Average Power
e Ie∗ = P + jQ
For any impedance, Z , complex power absorbed: S = V
 2  2 e |2
• Cosine Wave RMS
e = IZ     |
e (b) Ie × Ie∗ = Ie we get S = Ie Z = ∗V
• Power Factor Using (a) V Z
• Complex Power
• Power in R, L, C
 2 2
• Tellegen’s Theorem
 e |Ve |
• Power Factor Correction Resistor: S = I  R = R φ=0
• Ideal Transformer
• Transformer Applications
• Summary Absorbs average power, no VARs (Q = 0)

 2 2
 e |Ve |
Inductor: S = j I  ωL = j ωL φ = +90◦

No average power, Absorbs VARs (Q > 0)

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 Power in R, L, C

14: Power in AC Circuits

• Average Power
e Ie∗ = P + jQ
For any impedance, Z , complex power absorbed: S = V
 2  2 e |2
• Cosine Wave RMS
e = IZ     |
e (b) Ie × Ie∗ = Ie we get S = Ie Z = ∗V
• Power Factor Using (a) V Z
• Complex Power
• Power in R, L, C
 2 2
• Tellegen’s Theorem
 e |Ve |
• Power Factor Correction Resistor: S = I  R = R φ=0
• Ideal Transformer
• Transformer Applications
• Summary Absorbs average power, no VARs (Q = 0)

 2 2
 e |Ve |
Inductor: S = j I  ωL = j ωL φ = +90◦

No average power, Absorbs VARs (Q > 0)

2  2
|Ie|  
Capacitor: S = −j ωC = −j Ve  ωC φ = −90◦

No average power, Generates VARs (Q < 0)

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 Power in R, L, C

14: Power in AC Circuits

• Average Power
e Ie∗ = P + jQ
For any impedance, Z , complex power absorbed: S = V
 2  2 e |2
• Cosine Wave RMS
e = IZ     |
e (b) Ie × Ie∗ = Ie we get S = Ie Z = ∗V
• Power Factor Using (a) V Z
• Complex Power
• Power in R, L, C
 2 2
• Tellegen’s Theorem
 e |Ve |
• Power Factor Correction Resistor: S = I  R = R φ=0
• Ideal Transformer
• Transformer Applications
• Summary Absorbs average power, no VARs (Q = 0)

 2 2
 e |Ve |
Inductor: S = j I  ωL = j ωL φ = +90◦

No average power, Absorbs VARs (Q > 0)

2  2
|Ie|  
Capacitor: S = −j ωC = −j Ve  ωC φ = −90◦

No average power, Generates VARs (Q < 0)

VARs are generated by capacitors and absorbed by inductors

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 Power in R, L, C

14: Power in AC Circuits

• Average Power
e Ie∗ = P + jQ
For any impedance, Z , complex power absorbed: S = V
 2  2 e |2
• Cosine Wave RMS
e = IZ     |
e (b) Ie × Ie∗ = Ie we get S = Ie Z = ∗V
• Power Factor Using (a) V Z
• Complex Power
• Power in R, L, C
 2 2
• Tellegen’s Theorem
 e |Ve |
• Power Factor Correction Resistor: S = I  R = R φ=0
• Ideal Transformer
• Transformer Applications
• Summary Absorbs average power, no VARs (Q = 0)

 2 2
 e |Ve |
Inductor: S = j I  ωL = j ωL φ = +90◦

No average power, Absorbs VARs (Q > 0)

2  2
|Ie|  
Capacitor: S = −j ωC = −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
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 Tellegen’s Theorem

14: Power in AC Circuits

• Average Power
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s
• Cosine Wave RMS components sums to zero.
• Power Factor
• Complex Power
• Power in R, L, C
xn = voltage at node n
• Tellegen’s Theorem Vb , Ib = voltage/current in branch b
• Power Factor Correction
• Ideal Transformer (obeying passive sign convention)
• Transformer Applications
• Summary

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 Tellegen’s Theorem

14: Power in AC Circuits

• Average Power
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s
• Cosine Wave RMS components sums to zero.
• Power Factor
• Complex Power
• Power in R, L, C
xn = voltage at node n
• Tellegen’s Theorem Vb , Ib = voltage/current in branch b
• Power Factor Correction
• Ideal Transformer (obeying passive sign convention)
• Transformer Applications 
• Summary 
−1 if Vb starts from node n
abn , +1 if Vb ends at node n


0 else

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 Tellegen’s Theorem

14: Power in AC Circuits

• Average Power
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s
• Cosine Wave RMS components sums to zero.
• Power Factor
• Complex Power
• Power in R, L, C
xn = voltage at node n
• Tellegen’s Theorem Vb , Ib = voltage/current in branch b
• Power Factor Correction
• Ideal Transformer (obeying passive sign convention)
• Transformer Applications 
• Summary 
−1 if Vb starts from node n
abn , +1 if Vb ends at node n


0 else
e.g. branch 4 goes from 2 to 3 ⇒ a4∗ = [0, −1, 1]

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 Tellegen’s Theorem

14: Power in AC Circuits

• Average Power
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s
• Cosine Wave RMS components sums to zero.
• Power Factor
• Complex Power
• Power in R, L, C
xn = voltage at node n
• Tellegen’s Theorem Vb , Ib = voltage/current in branch b
• Power Factor Correction
• Ideal Transformer (obeying passive sign convention)
• Transformer Applications 
• Summary 
−1 if Vb starts from node n
abn , +1 if Vb ends at node n


0 else
e.g. branch 4 goes from 2 to 3 ⇒ a4∗ = [0, −1, 1]
P
Branch voltages: Vb = n abn xn (e.g. V4 = x3 − x2 )

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 Tellegen’s Theorem

14: Power in AC Circuits

• Average Power
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s
• Cosine Wave RMS components sums to zero.
• Power Factor
• Complex Power
• Power in R, L, C
xn = voltage at node n
• Tellegen’s Theorem Vb , Ib = voltage/current in branch b
• Power Factor Correction
• Ideal Transformer (obeying passive sign convention)
• Transformer Applications 
• Summary 
−1 if Vb starts from node n
abn , +1 if Vb ends at node n


0 else
e.g. branch 4 goes from 2 to 3 ⇒ a4∗ = [0, −1, 1]
P
Branch voltages: Vb = abn xn (e.g. V4 = x3 − x2 )
n
P P ∗
KCL @ node n: b abn Ib = 0 ⇒ b abn Ib = 0

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 Tellegen’s Theorem

14: Power in AC Circuits

• Average Power
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s
• Cosine Wave RMS components sums to zero.
• Power Factor
• Complex Power
• Power in R, L, C
xn = voltage at node n
• Tellegen’s Theorem Vb , Ib = voltage/current in branch b
• Power Factor Correction
• Ideal Transformer (obeying passive sign convention)
• Transformer Applications 
• Summary 
−1 if Vb starts from node n
abn , +1 if Vb ends at node n


0 else
e.g. branch 4 goes from 2 to 3 ⇒ a4∗ = [0, −1, 1]
P
Branch voltages: Vb = n 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

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 Tellegen’s Theorem

14: Power in AC Circuits

• Average Power
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s
• Cosine Wave RMS components sums to zero.
• Power Factor
• Complex Power
• Power in R, L, C
xn = voltage at node n
• Tellegen’s Theorem Vb , Ib = voltage/current in branch b
• Power Factor Correction
• Ideal Transformer (obeying passive sign convention)
• Transformer Applications 
• Summary 
−1 if Vb starts from node n
abn , +1 if Vb ends at node n


0 else
e.g. branch 4 goes from 2 to 3 ⇒ a4∗ = [0, −1, 1]
P
Branch voltages: Vb = n 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

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 Tellegen’s Theorem

14: Power in AC Circuits

• Average Power
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s
• Cosine Wave RMS components sums to zero.
• Power Factor
• Complex Power
• Power in R, L, C
xn = voltage at node n
• Tellegen’s Theorem Vb , Ib = voltage/current in branch b
• Power Factor Correction
• Ideal Transformer (obeying passive sign convention)
• Transformer Applications 
• Summary 
−1 if Vb starts from node n
abn , +1 if Vb ends at node n


0 else
e.g. branch 4 goes from 2 to 3 ⇒ a4∗ = [0, −1, 1]
P
Branch voltages: Vb = n 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∗
∗

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 Tellegen’s Theorem

14: Power in AC Circuits

• Average Power
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s
• Cosine Wave RMS components sums to zero.
• Power Factor
• Complex Power
• Power in R, L, C
xn = voltage at node n
• Tellegen’s Theorem Vb , Ib = voltage/current in branch b
• Power Factor Correction
• Ideal Transformer (obeying passive sign convention)
• Transformer Applications 
• Summary 
−1 if Vb starts from node n
abn , +1 if Vb ends at node n


0 else
e.g. branch 4 goes from 2 to 3 ⇒ a4∗ = [0, −1, 1]
P
Branch voltages: Vb = n 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

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 Tellegen’s Theorem

14: Power in AC Circuits

• Average Power
Tellegen’s Theorem: The complex power, S , dissipated in any circuit’s
• Cosine Wave RMS components sums to zero.
• Power Factor
• Complex Power
• Power in R, L, C
xn = voltage at node n
• Tellegen’s Theorem Vb , Ib = voltage/current in branch b
• Power Factor Correction
• Ideal Transformer (obeying passive sign convention)
• Transformer Applications 
• Summary 
−1 if Vb starts from node n
abn , +1 if Vb ends at node n


0 else
e.g. branch 4 goes from 2 to 3 ⇒ a4∗ = [0, −1, 1]
P
Branch voltages: Vb = n 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: b Sb = 0 ⇒ b Pb = 0 and also b Qb = 0.

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA

• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary
Add parallel capacitor of 300 µF:

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary
Add parallel capacitor of 300 µF:
1
ZC = jωC = −10.6j Ω

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary
Add parallel capacitor of 300 µF:
ZC = 1
jωC = −10.6j Ω ⇒ IeC = 21.7j A

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary
Add parallel capacitor of 300 µF:
ZC = jωC1
= −10.6j Ω ⇒ IeC = 21.7j A
Ie = 46 − j11.2 A = 47∠ − 14◦ A

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary
Add parallel capacitor of 300 µF:
ZC = jωC1
= −10.6j Ω ⇒ IeC = 21.7j A
Ie = 46 − j11.2 A = 47∠ − 14◦ A

SC = Ve IeC∗ = −j5 kVA

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary
Add parallel capacitor of 300 µF:
ZC = jωC1
= −10.6j Ω ⇒ IeC = 21.7j A
Ie = 46 − j11.2 A = 47∠ − 14◦ A

SC = Ve IeC∗ = −j5 kVA

S = Ve Ie∗ = 10.6 + j2.6 kVA

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary
Add parallel capacitor of 300 µF:
ZC = jωC1
= −10.6j Ω ⇒ IeC = 21.7j A
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

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary
Add parallel capacitor of 300 µF:
ZC = jωC1
= −10.6j Ω ⇒ IeC = 21.7j A
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

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary
Add parallel capacitor of 300 µF:
ZC = jωC1
= −10.6j Ω ⇒ IeC = 21.7j A
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.

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary
Add parallel capacitor of 300 µF:
ZC = jωC1
= −10.6j Ω ⇒ IeC = 21.7j A
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.
 e
I , reduced from 56.5 ց 47 A (−16%) ⇒ lower losses.

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary
Add parallel capacitor of 300 µF:
ZC = jωC1
= −10.6j Ω ⇒ IeC = 21.7j A
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.
 e
I , reduced from 56.5 ց 47 A (−16%) ⇒ lower losses.
Effect of C : VARs = 7.6 ց 2.6 kVAR

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 Power Factor Correction

Ve = 230. Motor modelled as 5||7j Ω.

14: Power in AC Circuits
• Average Power
e e
Ie = VR + ZVL = 46 − j32.9 A = 56.5∠ − 36◦
• Cosine Wave RMS
• Power Factor

S = Ve Ie∗ = 10.6 + j7.6 kVA = 13∠36◦ kVA

• Complex Power
• Power in R, L, C
P
• Tellegen’s Theorem
• Power Factor Correction
cos φ = |S| = cos 36◦ = 0.81
• Ideal Transformer
• Transformer Applications
• Summary
Add parallel capacitor of 300 µF:
ZC = jωC1
= −10.6j Ω ⇒ IeC = 21.7j A
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.
 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.

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 Ideal Transformer

14: Power in AC Circuits

• Average Power
A transformer has ≥ 2 windings on the same magnetic
• Cosine Wave RMS core.
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Ideal Transformer

14: Power in AC Circuits

• Average Power
A transformer has ≥ 2 windings on the same magnetic
• Cosine Wave RMS core.
• Power Factor
P lΦ Vr dΦ
• Complex Power
• Power in R, L, C
Ampère’s law: Nr Ir = µA ; Faraday’s law: N
r
= dt .
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Ideal Transformer

14: Power in AC Circuits

• Average Power
A transformer has ≥ 2 windings on the same magnetic
• Cosine Wave RMS core.
• Power Factor
P lΦ Vr dΦ
• Complex Power
• Power in R, L, C
Ampère’s law: Nr Ir = µA ;
Faraday’s law: N
r
= dt .
• Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings.
• Power Factor Correction
• Ideal Transformer The • indicates the voltage polarity of each winding.
• Transformer Applications
• Summary

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 Ideal Transformer

14: Power in AC Circuits

• Average Power
A transformer has ≥ 2 windings on the same magnetic
• Cosine Wave RMS core.
• Power Factor
P lΦ Vr dΦ
• Complex Power
• Power in R, L, C
Ampère’s law: Nr Ir = µA ;
Faraday’s law: N
r
= dt .
• Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings.
• Power Factor Correction
• Ideal Transformer The • indicates the voltage polarity of each winding.
• Transformer Applications
V1 V2 V3
• Summary Since Φ is the same for all windings, N = N2 = N3 .
1

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 Ideal Transformer

14: Power in AC Circuits

• Average Power
A transformer has ≥ 2 windings on the same magnetic
• Cosine Wave RMS core.
• Power Factor
P lΦ Vr dΦ
• Complex Power
• Power in R, L, C
Ampère’s law: Nr Ir = µA ;
Faraday’s law: N
r
= dt .
• Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings.
• Power Factor Correction
• Ideal Transformer The • indicates the voltage polarity of each winding.
• Transformer Applications
V1 V2 V3
• Summary Since Φ is the same for all windings, N = N2 = N3 .
1

Assume µ → ∞ ⇒ N1 I1 + N2 I2 + N3 I3 = 0

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 Ideal Transformer

14: Power in AC Circuits

• Average Power
A transformer has ≥ 2 windings on the same magnetic
• Cosine Wave RMS core.
• Power Factor
P lΦ Vr dΦ
• Complex Power
• Power in R, L, C
Ampère’s law: Nr Ir = µA ;
Faraday’s law: N
r
= dt .
• Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings.
• Power Factor Correction
• Ideal Transformer The • indicates the voltage polarity of each winding.
• Transformer Applications
V1 V2 V3
• Summary Since 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.

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 Ideal Transformer

14: Power in AC Circuits

• Average Power
A transformer has ≥ 2 windings on the same magnetic
• Cosine Wave RMS core.
• Power Factor
P lΦ Vr dΦ
• Complex Power
• Power in R, L, C
Ampère’s law: Nr Ir = µA ;
Faraday’s law: N
r
= dt .
• Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings.
• Power Factor Correction
• Ideal Transformer The • indicates the voltage polarity of each winding.
• Transformer Applications
V1 V2 V3
• Summary Since 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:

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 Ideal Transformer

14: Power in AC Circuits

• Average Power
A transformer has ≥ 2 windings on the same magnetic
• Cosine Wave RMS core.
• Power Factor
P lΦ Vr dΦ
• Complex Power
• Power in R, L, C
Ampère’s law: Nr Ir = µA ;
Faraday’s law: N
r
= dt .
• Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings.
• Power Factor Correction
• Ideal Transformer The • indicates the voltage polarity of each winding.
• Transformer Applications
V1 V2 V3
• Summary Since 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
1
I12

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 Ideal Transformer

14: Power in AC Circuits

• Average Power
A transformer has ≥ 2 windings on the same magnetic
• Cosine Wave RMS core.
• Power Factor
P lΦ Vr dΦ
• Complex Power
• Power in R, L, C
Ampère’s law: Nr Ir = µA ;
Faraday’s law: N
r
= dt .
• Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings.
• Power Factor Correction
• Ideal Transformer The • indicates the voltage polarity of each winding.
• Transformer Applications
V1 V2 V3
• Summary Since 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
1
I1 2
 2  2
Hence VI 1 = N1
N2
V2
IL = N1
N2 Z
1

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 Ideal Transformer

14: Power in AC Circuits

• Average Power
A transformer has ≥ 2 windings on the same magnetic
• Cosine Wave RMS core.
• Power Factor
P lΦ Vr dΦ
• Complex Power
• Power in R, L, C
Ampère’s law: Nr Ir = µA ;
Faraday’s law: N
r
= dt .
• Tellegen’s Theorem N1 : N2 + N3 shows the turns ratio between the windings.
• Power Factor Correction
• Ideal Transformer The • indicates the voltage polarity of each winding.
• Transformer Applications
V1 V2 V3
• Summary Since 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
1
I1 2
 2  2
Hence VI 1 = N1
N2
V2
IL = N1
N2 Z
1
 2
N1
Equivalent to a reflected impedance of N2 Z
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 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

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 Transformer Applications

14: Power in AC Circuits

Power Transmission
• Average Power
• Cosine Wave RMS
• Power Factor Suppose a power transmission cable has 1 Ω resistance.
• Complex Power
• Power in R, L, C
100 kVA@ 1 kV = 100 A ⇒ Ie2 R = 10 kW losses.
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Transformer Applications

14: Power in AC Circuits

Power Transmission
• Average Power
• Cosine Wave RMS
• Power Factor Suppose a power transmission cable has 1 Ω resistance.
• Complex Power
• Power in R, L, C
100 kVA@ 1 kV = 100 A ⇒ Ie2 R = 10 kW losses.
• Tellegen’s Theorem 100 kVA@ 100 kV = 1 A ⇒ Ie2 R = 1 W losses.
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Transformer Applications

14: Power in AC Circuits

Power Transmission
• Average Power
• Cosine Wave RMS
• Power Factor Suppose a power transmission cable has 1 Ω resistance.
• Complex Power
• Power in R, L, C
100 kVA@ 1 kV = 100 A ⇒ Ie2 R = 10 kW losses.
• Tellegen’s Theorem 100 kVA@ 100 kV = 1 A ⇒ Ie2 R = 1 W losses.
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
Voltage Conversion
• Summary
Electronic equipment requires ≤ 20 V but mains voltage is 240 V ∼.

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 Transformer Applications

14: Power in AC Circuits

Power Transmission
• Average Power
• Cosine Wave RMS
• Power Factor Suppose a power transmission cable has 1 Ω resistance.
• Complex Power
• Power in R, L, C
100 kVA@ 1 kV = 100 A ⇒ Ie2 R = 10 kW losses.
• Tellegen’s Theorem 100 kVA@ 100 kV = 1 A ⇒ Ie2 R = 1 W losses.
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
Voltage Conversion
• Summary
Electronic equipment requires ≤ 20 V but mains voltage is 240 V ∼.

Interference protection

Microphone on long cable is susceptible to interference from nearby

mains cables. An N : 1 transformer reduces the microphone voltage
by N but reduces interference by N 2 .

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 Transformer Applications

14: Power in AC Circuits

Power Transmission
• Average Power
• Cosine Wave RMS
• Power Factor Suppose a power transmission cable has 1 Ω resistance.
• Complex Power
• Power in R, L, C
100 kVA@ 1 kV = 100 A ⇒ Ie2 R = 10 kW losses.
• Tellegen’s Theorem 100 kVA@ 100 kV = 1 A ⇒ Ie2 R = 1 W losses.
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
Voltage Conversion
• Summary
Electronic equipment requires ≤ 20 V but mains voltage is 240 V ∼.

Interference protection

Microphone on long cable is susceptible to interference from nearby

mains cables. An N : 1 transformer reduces the microphone voltage
by N but reduces interference by N 2 .

Isolation

There is no electrical connection between the windings of a transformer

so circuitry (or people) on one side will not be endangered by a failure
that results in high voltages on the other side.

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 Summary

• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms =

14: Power in AC Circuits
• Average Power √1 V .
2
• Cosine Wave RMS
• Power Factor
• Complex Power
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Summary

• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms =

14: Power in AC Circuits
• Average Power √1 V .
2
• Cosine Wave RMS  2 2
• Power Factor  e |Ve |
• Complex Power ◦ For an impedance Z : S = I  Z = Z ∗
• Power in R, L, C
• Tellegen’s Theorem
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Summary

• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V .

14: Power in AC Circuits
• Average Power
• Cosine Wave RMS  2 2
• Power Factor  e |Ve |
• Complex Power ◦ For an impedance Z : S = I  Z = Z ∗
  
• Power in R, L, C
  
• Tellegen’s Theorem ◦ Apparent Power: |S| = Ve  Ie used for machine ratings.
• Power Factor Correction
• Ideal Transformer
• Transformer Applications
• Summary

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 Summary

• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V .

14: Power in AC Circuits
• Average Power
• Cosine Wave RMS  2 2
• Power Factor  e |Ve |
• Complex Power ◦ For an impedance Z : S = I  Z = Z ∗
  
• Power in R, L, C
  
• Tellegen’s Theorem ◦ Apparent Power: |S| = Ve  Ie used for machine ratings.
• Power Factor Correction   
  
◦ Average Power: P = ℜ (S) = Ve  Ie cos φ (in Watts)
• Ideal Transformer
• Transformer Applications
• Summary   
  
◦ Reactive Power: Q = ℑ (S) = Ve  Ie sin φ (in VARs)

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 Summary

• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V .

14: Power in AC Circuits
• Average Power
• Cosine Wave RMS  2 2
• Power Factor  e |Ve |
• Complex Power ◦ For an impedance Z : S = I  Z = Z ∗
  
• Power in R, L, C
  
• Tellegen’s Theorem ◦ Apparent Power: |S| = Ve  Ie used for machine ratings.
• Power Factor Correction   
  
◦ Average Power: P = ℜ (S) = Ve  Ie cos φ (in Watts)
• Ideal Transformer
• Transformer Applications
• Summary   
  
◦ Reactive Power: Q = ℑ (S) = Ve  Ie sin φ (in VARs)
◦ Power engineers always use Ve and Ie and omit the ~.

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 Summary

• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V .

14: Power in AC Circuits
• Average Power
• Cosine Wave RMS  2 2
• Power Factor  e |Ve |
• Complex Power ◦ For an impedance Z : S = I  Z = Z ∗
  
• Power in R, L, C
  
• Tellegen’s Theorem ◦ Apparent Power: |S| = Ve  Ie used for machine ratings.
• Power Factor Correction   
  
◦ Average Power: P = ℜ (S) = Ve  Ie cos φ (in Watts)
• Ideal Transformer
• Transformer Applications
• Summary   
  
◦ Reactive Power: Q = ℑ (S) = Ve  Ie sin φ (in VARs)
◦ Power engineers always use Ve and Ie and omit the ~.
P P P
• Tellegen: In any circuit b Sb = 0 ⇒ b Pb = b Qb = 0

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 Summary

• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V .

14: Power in AC Circuits
• Average Power
• Cosine Wave RMS  2 2
• Power Factor  e |Ve |
• Complex Power ◦ For an impedance Z : S = I  Z = Z ∗
  
• Power in R, L, C
  
• Tellegen’s Theorem ◦ Apparent Power: |S| = Ve  Ie used for machine ratings.
• Power Factor Correction   
  
◦ Average Power: P = ℜ (S) = Ve  Ie cos φ (in Watts)
• Ideal Transformer
• Transformer Applications
• Summary   
  
◦ Reactive Power: Q = ℑ (S) = Ve  Ie sin φ (in VARs)
◦ Power engineers always use Ve and Ie and omit the ~.
P P P
• Tellegen: In any circuit b Sb = 0 ⇒ b Pb = b Qb = 0

• Power Factor Correction: add parallel C to generate extra VARs

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 Summary

• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V .

14: Power in AC Circuits
• Average Power
• Cosine Wave RMS  2 2
• Power Factor  e |Ve |
• Complex Power ◦ For an impedance Z : S = I  Z = Z ∗
  
• Power in R, L, C
  
• Tellegen’s Theorem ◦ Apparent Power: |S| = Ve  Ie used for machine ratings.
• Power Factor Correction   
  
◦ Average Power: P = ℜ (S) = Ve  Ie cos φ (in Watts)
• Ideal Transformer
• Transformer Applications
• Summary   
  
◦ Reactive Power: Q = ℑ (S) = Ve  Ie sin φ (in VARs)
◦ Power engineers always use Ve and Ie and omit the ~.
P P P
• Tellegen: In any circuit b Sb = 0 ⇒ b Pb = b Qb = 0

• Power Factor Correction: add parallel C to generate extra VARs

P P
• Ideal Transformer: Vi ∝ Ni and Ni Ii = 0 (implies Si = 0)

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 Summary

• Complex Power: S = Ve Ie∗ = P + jQ where Ve = Vrms = √12 V .

14: Power in AC Circuits
• Average Power
• Cosine Wave RMS  2 2
• Power Factor  e |Ve |
• Complex Power ◦ For an impedance Z : S = I  Z = Z ∗
  
• Power in R, L, C
  
• Tellegen’s Theorem ◦ Apparent Power: |S| = Ve  Ie used for machine ratings.
• Power Factor Correction   
  
◦ Average Power: P = ℜ (S) = Ve  Ie cos φ (in Watts)
• Ideal Transformer
• Transformer Applications
• Summary   
  
◦ Reactive Power: Q = ℑ (S) = Ve  Ie sin φ (in VARs)
◦ Power engineers always use Ve and Ie and omit the ~.
P P P
• Tellegen: In any circuit b Sb = 0 ⇒ b Pb = b Qb = 0

• Power Factor Correction: add parallel C to generate extra VARs

P P
• Ideal Transformer: Vi ∝ Ni and Ni Ii = 0 (implies Si = 0)

For further details see Hayt et al. Chapter 11.

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