instantaneous and 3-phase power - chatziva

Spyros Chatzivasileiadis

31730: Electric Power Engineering, fundamentals

Instantaneous and 3-phase power

8 September 2017

If not otherwise indicated, all figures are taken from the course textbook: J. D. Glover, T. J. Overbye, M. S. Sarma, Power System Analysis and Design, Sixth Edition - SI, Cengage Learning, 2016

DTU Electrical Engineering, Technical University of Denmark

Reviewing previous lecture

2

• For 5 minutes discuss with the person sitting next to you about:

– Three main points we discussed in the last lecture

– One topic or concept that is not so clear and you would like to hear again about it



The goals for today• Instantaneous active and reactive power

• Power factor and complex power

• Line-to-neutral and line-to-line voltage

• Δ-loads and Y-loads

• Single-line diagram

• Power in three-phase balanced systems

3 Spyros Chatzivasileiadis


Instantaneous power• On the board



Phasor Diagrams

5

Z

Re

Im

Z=R

Z=jXC

Z=jXL

IIm

I=IR

I=IL

I=Ic

PowerIm

Q=PR

S=jQC

S=jQL

Re Re

• Inductive Character:– Current lags the voltage– Q consumed; Q>0 (for loads)

• Capacitive Character:– Current leads the voltage– Q produced; Q<0 (for loads)



Conventions: Positive and negative P and Q



Balanced three-phase systems

7

• Voltage line-to-neutral

• Voltage line-to-line

• Question:What are the phasors for the line to neutral voltages?(for each phase)



Balanced three-phase systems

8

• Voltage line-to-neutral

• Voltage line-to-line

• Assuming a 𝑉"#$ = 10for each phase:

• Question:What are the phasors for the line-to-line voltages?(for each phase)

S = V · I∗

ZY =Z∆

3

p3φ(t) = P3φ = 3VLNIL cos(δ − β)

Ean = 10 0◦

Ebn = 10 −120◦

Ecn = 10 +120◦

7 DTU Electrical Engineering 31730: Electric Power Engineering, Fundamentals Sep 8, 2017



Line-to-neutral and line-to-line voltages



Delta-Wye Loads and transformation

10

S = V · I∗

ZY =Z∆

3

p3φ(t) = pa(t) + pb(t) + pc(t) =

3VLNIL cos(δ − β) + VLNIL[cos(2ωt+ δ + β)

cos(2ωt+ δ + β − 240◦)+ cos(2ωt+ δ + β + 240◦)]




Three-phase to single line diagram



Three-phase to single line diagram

12

• Three main points:

1. Voltages must be RMS values

2. Transform line-to-line voltages to line-to-neutral

3. Transform all Δ-loads to Y-loads



Power balanced in three-phase systems

• The three-phase instantaneous power remains constant!

• Relationship between line-to-line and line-to-neutral differs by √3

• Similar for Q

13

S = V · I∗

ZY =Z∆

3

p3φ(t) =pa(t) + pb(t) + pc(t) =

3VLNIL cos(δ − β) + VLNIL[cos(2ωt+ δ + β)+

cos(2ωt+ δ + β − 240◦)+

cos(2ωt+ δ + β + 240◦)]


p3φ(t) = P3φ =√3VLLIL cos(δ − β)


S = V · I∗

ZY =Z∆

3

p3φ(t) =pa(t) + pb(t) + pc(t) =


cos(2ωt+ δ + β − 240◦)+

cos(2ωt+ δ + β + 240◦)]




S = V · I∗

ZY =Z∆

3

p3φ(t) =pa(t) + pb(t) + pc(t) =


cos(2ωt+ δ + β − 240◦)+

cos(2ωt+ δ + β + 240◦)]




S = V · I∗

ZY =Z∆

3



p3φ(t) = Q3φ = 3VLNIL sin(δ − β)




Why did three-phase systems prevail?



Three-phase vs. single-phase systems

15

• Benefits

1. Neutral current and voltage is zeroa. need for less conductorsb. Lower costs for transmission

2. Total instantaneous power remains constanta. Single-phase would create shaft

vibration and noise in motors and generators

b. Shaft failures in large generators c. Gens>5 kVA are three-phase


instantaneous and 3-phase power - chatziva

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