Electric Circuits (Fall 2015) Pingqiang Zhou
Lecture 12
- Three-Phase Circuits/Transformers
Part I
11/26/2015
Reading: Chapter 12
1Lecture 12
Electric Circuits (Fall 2015) Pingqiang Zhou
Outline
• Why Three-Phase for AC supply?
• Balanced Three-Phase System
Balanced sources
Balanced loads
• Circuit analysis
Phase voltage/current
Line voltage/current
Lecture 12 2
Electric Circuits (Fall 2015) Pingqiang Zhou
Two-Pin and Three-Pin Sockets
3Lecture 12
https://en.wikipedia.org/wiki/AC_power_plugs_and_sockets
Electric Circuits (Fall 2015) Pingqiang Zhou
Electrical Safety
4Lecture 12
https://cnx.org/contents/8f205833-26b8-4eb4-98bb-936aad728cc6@2/Electrical-Safety-Systems-and-
Electric Circuits (Fall 2015) Pingqiang Zhou
5Lecture 12
Electric Circuits (Fall 2015) Pingqiang Zhou
Single Phase vs. Polyphase
• Households have single-phase power supply
This typically in a three wire form, where two 120V sources with the
same phase are connected in series.
This allows for appliances to use either 120 or 240V
• Circuits that operate at the same frequency but with multiple
sources at different phases are called polyphase.
Lecture 12 6
Electric Circuits (Fall 2015) Pingqiang Zhou
Three-Phase System
• In power grids, three phase
power is used for a variety of
reasons.
It is easy to extract single or two
phase power from a three phase
system, satisfying the cases where
this is needed.
The instantaneous power in a three
phase system does not pulsate like
it does in a single phase system.
(refer to Ch. 12.7)
Lastly, the transmission of three
phase is more economical than
transmitting the equivalent single
phase power. (refer to Ch. 12.7)
Lecture 12 7
Electric Circuits (Fall 2015) Pingqiang Zhou
Balanced Three Phase
• Three phase voltages are typically
produced by a three phase AC
generator.
• The output voltages look like below.
Lecture 12 8
Electric Circuits (Fall 2015) Pingqiang Zhou
Connecting the Sources
• Three phase voltage sources can be connected the loads
by either three or four wire configurations.
Three-wire configuration accomplished by Delta connected source.
Four-wire system accomplished using a Y connected source.
Lecture 12 9
Electric Circuits (Fall 2015) Pingqiang Zhou
Balanced Source
• A wye connected source is said
to be balanced when
• Two sequences for the phases:
0an bn cnV V V
an bn cnV V V
0
120
240 120
an p
bn p
cn p p
V V
V V
V V V
0
120
240 120
an p
cn p
bn p p
V V
V V
V V V
Lecture 12 10
Positive Negative
Electric Circuits (Fall 2015) Pingqiang Zhou
Balanced Loads
• Similar to the source, a balanced load is one that has the
same impedance presented to all three voltage sources.
• They may also be connected in either Delta or wye
For a balanced wye connected load:
For a balanced delta connected load:1 2 3 YZ Z Z Z
a b cZ Z Z Z
Lecture 12 11
Electric Circuits (Fall 2015) Pingqiang Zhou
Source-Load configurations
• The load impedance per phase for the two load
configurations can be interchanged:
Lecture 12 12
Electric Circuits (Fall 2015) Pingqiang Zhou
Y and ∆, Which One Better?
13Lecture 12
http://www.allaboutcircuits.com/textbook/alternating-current/chpt-10/three-phase-y-delta-configurations/
Electric Circuits (Fall 2015) Pingqiang Zhou
Balanced Y-Y connection
• Any three-phase system can
be reduced to an equivalent
Y-Y system.
• We will consider an example
of a balanced four wire Y-Y
system shown here.
• The load impedances Zy will
be assumed to be balanced.
This can be the source 𝑍𝑠, line
𝑍𝑙 and load 𝑍𝐿 together.
Lecture 12 14
Electric Circuits (Fall 2015) Pingqiang Zhou
Line-to-Line Voltage
• We will use the positive sequence for
this circuit, meaning the voltages are:
0
120 120
an p
bn p cn p
V V
V V V V
The line to line (or line in short) voltages:Thus the magnitude of the line
voltages VL is:
3L pV V
p an bn cn
L ab bc ca
V V V V
V V V V
3 30
3 90
3 210
ab P
bc P
ca P
V V
V V
V V
Lecture 12 15
Electric Circuits (Fall 2015) Pingqiang Zhou
Line Currents
• If we apply KVL to each phase, we
find the line currents are:
120
240
ana b a
Y
c a
VI I I
Z
I I
0a b cI I I
• From this one can see the line currents add up to zero.
This shows that the neutral wire has zero voltage and no current.
Thus it can be removed without affecting the system.
Lecture 12 16
Electric Circuits (Fall 2015) Pingqiang Zhou
Per Phase Analysis
• An alternative way to analyze the
Y-Y circuit is to look at each phase
individually. Let us look at phase a:
The equivalent circuit for that phase is
shown here.
The current for this phase is:
• If the circuit is balanced, only one
phase need be analyzed.
ana
Y
VI
Z
Lecture 12 17
Electric Circuits (Fall 2015) Pingqiang Zhou
Example
• Calculate the line currents.
18Lecture 12
Electric Circuits (Fall 2015) Pingqiang Zhou
Balanced Wye-Delta
• This system consists of a
balanced Y connected source and
a balanced Delta connected load.0
120 120
an p
bn p cn p
V V
V V V V
3 30 3 90 3 150ab P AB bc P BC ca P CAV V V V V V V V V
The line voltages are equal to the voltages across the
load. From this, we can calculate the phase currents:
The line voltages are:
BC CAABAB BC CA
V VVI I I
Z Z Z
Lecture 12 19
Electric Circuits (Fall 2015) Pingqiang Zhou
Balanced Wye-Delta II
• An alternative way to solve for the
phase currents is to apply KVL.
• For example, applying KVL
around the loop aABbna gives:
• Or
• This is the more general way to
find phase currents.
0an AB bnV Z I V
an bn ab ABAB
V V V VI
Z Z Z
Lecture 12 20
Electric Circuits (Fall 2015) Pingqiang Zhou
Phase to Line Currents
• The line currents can be obtained from the phase currents
by applying KCL to nodes A, B, and C
Since ICA = IAB -240˚:
Thus:
a AB CA b BC AB c CA BCI I I I I I I I I
3 30a ABI I
3L pI I
Lecture 12 21
Electric Circuits (Fall 2015) Pingqiang Zhou
Alternative
• An alternate way to analyze the
Wye-Delta circuit is to transform
the Delta connected load into a
wye connected load. Using the
Delta-Wye transformation:
3Y
ZZ
With this circuit now rendered as a Y-Y circuit, single phase
analysis can be done
Lecture 12 22
Electric Circuits (Fall 2015) Pingqiang Zhou
Balanced Delta-Delta
• Now we turn our attention to the Delta-Delta configuration.
• Once again, the goal is to get the phase and line currents.
• Note that Delta configured generators are less typical than
the wye because any imbalance in the voltage sources will
result in current flowing through the delta loop.
Lecture 12 23
Electric Circuits (Fall 2015) Pingqiang Zhou
Balanced Delta-Delta II
• Assuming a positive sequence,
the phase voltages are:
0
120 120
ab p
bc p ca p
V V
V V V V
ab AB bc BC ca CAV V V V V V
• If line impedances are insignificant, then the line voltages
are the same as the phase voltages.
Hence the phase currents are:
ab BC bc CA caABAB BC CA
V V V V VVI I I
Z Z Z Z Z Z
a AB CA b BC AB c CA BCI I I I I I I I I 3L pI I
Lecture 12 24
Electric Circuits (Fall 2015) Pingqiang Zhou
Balanced Delta-Wye
• The last configuration to consider is the Delta-Wye system.
• The phase voltages are the same as the last case.
• There are many ways to get the line currents.
• One way is to apply KVL to the loop aANNba
0Y a b ab pZ I I V V
0p
a b
Y
VI I
Z
Keeping in mind that Ib lags Ia by
120˚, we can solve for the line
current:
/ 3 30p
a
Y
VI
Z
Lecture 12 25
Electric Circuits (Fall 2015) Pingqiang Zhou
Convert back to Y-Y
• Another way to solve this system is to
convert both the source and load
back to a Wye-Wye system.
• The equivalent Wye connected
source voltages are:
303
150 903 3
p
an
p p
bn cn
VV
V VV V
The load conversion goes as the
standard delta-wye conversion. Once
this is done, a single phase can be
examined to find the line current.
/ 3 30p
a
Y
VI
Z
Lecture 12 26
Electric Circuits (Fall 2015) Pingqiang Zhou
Summary:
Lecture 12 27
Electric Circuits (Fall 2015) Pingqiang Zhou
Summary II:
Lecture 12 28
Electric Circuits (Fall 2015) Pingqiang Zhou
Power in a Balanced System
• Power in balanced Wye load
29
[Nilsson, Ch. 11.5]
Lecture 12
Electric Circuits (Fall 2015) Pingqiang Zhou
Power in a Balanced System
• Power in balanced Delta load
30Lecture 12
[Nilsson, Ch. 11.5]
Electric Circuits (Fall 2015) Pingqiang Zhou
Further Reading
• Unbalanced Three-Phase System (Ch.12.8)
• Two-wattmeter method (Ch.12.10.1)
• Residential wiring (Ch.12.10.2)
• http://www.allaboutcircuits.com/textbook/alternating-
current/#chpt-10
31Lecture 12
Electric Circuits (Fall 2015) Pingqiang Zhou
Lecture 12
- Three-Phase Circuits/Transformers
Part II
11/26/2015
Reading: Chapter 13
32Lecture 12
Electric Circuits (Fall 2015) Pingqiang Zhou
Outline
• Mutual inductance
• Transformers
33Lecture 12
Electric Circuits (Fall 2015) Pingqiang Zhou
34Lecture 12
Electric Circuits (Fall 2015) Pingqiang Zhou
Self Inductance and Mutual Inductance
• Self inductance: reaction of
the inductor to the change in
current through itself.
𝑣 = 𝑁𝑑𝜙
𝑑𝑡= 𝑁
𝑑𝜙
𝑑𝑖
𝑑𝑖
𝑑𝑡= 𝐿
𝑑𝑖
𝑑𝑡
𝐿 = 𝑁𝑑𝜙
𝑑𝑖
• Mutual inductance: reaction of
the inductor to change in current
through another inductor.
𝑣1 = 𝐿1𝑑𝑖1𝑑𝑡
𝐿1 = 𝑁1𝑑𝜙1𝑑𝑖1
𝑣2 = 𝑁2𝑑𝜙12𝑑𝑡
= 𝑁2𝑑𝜙12𝑑𝑖1
𝑑𝑖1𝑑𝑡
= 𝑀21
𝑑𝑖1𝑑𝑡
𝑀21 = 𝑁2𝑑𝜙12𝑑𝑖1
Lecture 12 35
Electric Circuits (Fall 2015) Pingqiang Zhou
Example
• Knowing the dot convention, we can analyze the series
aiding connection.
Applying KVL to coil 1:
For coil 2:
In the frequency domain:
1 21 1 1 1
di div i R L M
dt dt
2 12 2 2 2
di div i R L M
dt dt
𝐕𝟏 = 𝑅1 + 𝑗𝜔𝐿1 𝐈𝟏 + 𝑗𝜔𝑀𝐈𝟐
𝐕𝟐 = 𝑗𝜔𝑀𝐈𝟏 + 𝑅2 + 𝑗𝜔𝐿2 𝐈𝟐
Lecture 12 36
Electric Circuits (Fall 2015) Pingqiang Zhou
Energy in a Coupled Circuit
• The energy stored in an inductor is:
• For coupled inductors, the total
energy stored depends on the
individual inductance and on the
mutual inductance.
• The positive sign is selected when
the currents both enter or leave the
dotted terminals.
21
2w Li
2 2
1 1 2 2 1 2
1 1
2 2w L i L i Mi i
Lecture 12 37
Electric Circuits (Fall 2015) Pingqiang Zhou
Coupling Coefficient 𝒌
• With the total energy established for the mutual inductors,
we can establish an upper limit on M.
• The system cannot have negative energy because the
system is passive.
2 2
1 1 2 2 1 2
1 10
2 2L i L i Mi i
1 2M L L
• Define a parameter describes how
closely M approaches upper limit.
• Coupling coefficient, 0 ≤ 𝑘 ≤ 1.
• determined by the physical
configuration of the coils.
1 2
Mk
L L
Lecture 12 38
Electric Circuits (Fall 2015) Pingqiang Zhou
Linear Transformers
• A transformer is a magnetic device that takes advantage
of mutual inductance.
Generally a four terminal device comprised of two or more
magnetically coupled coils.
They are called linear if the coils are wound on a magnetically
linear material.
Lecture 12 39
Electric Circuits (Fall 2015) Pingqiang Zhou
Transformer Impedance
• An important parameter to know for a transformer is how
the input impedance Zin is seen from the source.
Zin is important because it governs the behavior of the primary
circuit.
Reflected impedance from
secondary to primary
Lecture 12 40
Electric Circuits (Fall 2015) Pingqiang Zhou
Equivalent circuits
• We already know that coupled inductors can be tricky to
work with.
• One approach is to use a transformation to create an
equivalent circuit.
The goal is to remove the mutual inductance.
This can be accomplished by using a T or a network.
The goal is to match the terminal voltages and currents from the
original network to the new network.
Lecture 12 41
Electric Circuits (Fall 2015) Pingqiang Zhou
Equivalent Circuits (T or )
1 1 1
2 2 2
V j L j M I
V j M j L I
Lecture 12 42
Electric Circuits (Fall 2015) Pingqiang Zhou
Ideal Transformers
• The ideal transformer has:
Coils with very large reactance
(L1, L2, M →)
Coupling coefficient k is equal to
unity.
Primary and secondary coils are
lossless
Lecture 12 43
Electric Circuits (Fall 2015) Pingqiang Zhou
Ideal Transformers II
• The voltages are related to each
other by the turns ratio n:
• The current is related as:
• Reflected impedance
1 2
2 1
V Nn
V N
2 1
1 2
1I N
I N n
Lecture 12 44
Electric Circuits (Fall 2015) Pingqiang Zhou
Ideal Autotransformer
• Autotransformer uses one winding for primary & secondary
It does not offer isolation!
Lecture 12 45
Electric Circuits (Fall 2015) Pingqiang Zhou
Three Phase Transformer
• Three-phase power has two choices for transformers:
A transformer bank, with one transformer per phase
A three phase transformer (smaller and less expensive)
Lecture 12 46
http://www.allaboutcircuits.com/textbook/alternating-current/chpt-10/three-phase-transformer-circuits/
Electric Circuits (Fall 2015) Pingqiang Zhou
Further Reading (Ch. 13.9)
• Transformer as isolation/matching device
47Lecture 12
• Power distribution
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