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7/17/2019 Components of symmetric fault analysis
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Topic 2
Symmetrical Components
0
Symmetrical Components
7/17/2019 Components of symmetric fault analysis
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Outline
• Concept and definition of symmetrical components
• Sequence networks of impedance loads
• Sequence networks of series impedances
•
Sequence networks of transmission and distribution lines• Sequence networks of rotating machines
• Sequence networks of transformers
• Power in sequence networks
• Main reference: Ch 8 Textbook
Symmetrical Components
1
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Outline
• Concept and definition of symmetrical components
• Sequence networks of impedance loads
• Sequence networks of series impedances
•
Sequence networks of transmission and distribution lines• Sequence networks of rotating machines
• Sequence networks of transformers
• Power in sequence networks
• Main reference: Ch 8 Textbook
Symmetrical Components
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Symmetric Components
Charles FortescueDeveloped symmetrical component analysis
Symmetrical Components
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Symmetric Components
Introduced to simplify calculation of unbalancednetworks, in particular networks with unbalanced short
circuits and other unbalanced fault conditions.
Symmetrical Components
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The Problem
Analysis of balanced three phase networks is simple
Analysis of unbalanced three phase networks is difficult
Solution
Adapt balanced analysis to use for unbalanced systems
Symmetrical Components
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The Principle
Any unbalanced set of three phase phasor quantities (V or I)
can be resolved into three balanced systems of phasor
quantities
Sequence components are the most commonly used set of
symmetrical components
Symmetrical Components
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Symmetric Components
• The key idea of symmetrical component analysis is todecompose the system into three sequence networks.
The networks are then coupled only at the point of the
unbalance (i.e., the fault)
•
The three sequence networks are known as the – positive sequence (this is the one we’ve been
using)
– negative sequence
– zero sequence
Symmetrical Components
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Positive Sequence Sets
• The positive sequence sets have three phase
currents/voltages with equal magnitude, with phase b
lagging phase a by 120°, and phase c lagging phase b by
120°.
• We’ve been studying positive sequence sets
Symmetrical Components
8
Positive sequence
sets have zero
neutral current
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Negative Sequence Sets
•
The negative sequence sets have three phasecurrents/voltages with equal magnitude, with phase b
leading phase a by 120°, and phase c leading phase b by
120°.
•
Negative sequence sets are similar to positive sequence,except the phase order is reversed
Symmetrical Components
9
Negative sequencesets have zero
neutral current
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Zero Sequence Sets
• Zero sequence sets have three values with equal magnitude
and angle.
• Zero sequence sets have neutral current
Symmetrical Components
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Symmetrical Components Combination
Symmetrical Components
11
Symmetrical Components combined to create a set of unbalanced
three phase phasor quantities
Ia1 Ia2
Ia0
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Sequence Set Representation
• Any arbitrary set of three phasors, say I a , I
b , I
c, can be
represented as a sum of the three sequence sets
Symmetrical Components
12
0 1 2
0 1 2
0 1 2
0 0 0
1 1 1
2 2 2
where
, , is the zero sequence set, , is the positive sequence set
, , is the negative sequence set
a a a a
b b b b
c c c c
a b c
a b c
a b c
I I I I
I I I I
I I I I
I I I I I I
I I I
S i l C
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Mathematical Operators
Symmetrical Components
13
2
3
2
3
2
1 90
1 120 0.5 0.866
1 240 1 1201 0
1 0
j
j
e j
2
1 = 3
S t i l C t
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Conversion from Sequence to Phase
Symmetrical Components
14
0 1 1
0 0 02 2
1 1 1 1 2 2 1 2
Only three of the sequence values are unique,
, , ; the others are determined as follows:
(since by definition they are all equal)
a a a
a b c
b a c a b a c a
a
b
c
I I I
I I I
I I I I I I I I
I
I
I
0
2 2a0 a1 2 1
2 22
1 1 1 11 1
I 1 I 1
1 1
a
a a
a
I
I I
I
S t i l C t
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Conversion from Sequence to Phase
Symmetrical Components
15
2
2
0 0
1 1
2 2
Define the symmetrical components transformation
matrix
1 1 1
1
1
Thenaa
b a s
c a
I I I I I I
I I I
A
I A A A I
S t i l C t
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Conversion from Sequence to Phase
Symmetrical Components
16
0 0
1 1
2 2
0 1 2
20 1
(removing subscribe )
Sequence vector consisting of sequence currents
Thus
a
s a
a
a
b
I I
I I a
I I
I I I I
I I I
I
2
20 1 2c
I
I I I I
Symmetrical Components
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Conversion from Phase to Sequence
Symmetrical Components
17
1
1 2
2
By taking the inverse we can convert from the
phase values to the sequence values
1 1 11
with
Sequence sets can be used with voltages as well
as with curr
1
en
1
ts
3
s
I A I
A
Symmetrical Components
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Conversion from Phase to Sequence
Symmetrical Components
18
1 1 2
2
0
21
22
1 1 11
with 1 implies3
1
( ) / 3
( ) / 3
( ) / 3
s
a b c
a b c
a b c
I I I I
I I I I
I I I I
I A I A
Symmetrical Components
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Neutral Currents in 3ph Systems
Symmetrical Components
19
0
0
0
In a 4 wire Y-connected system, 3 phases are connectedto the neutral and the neutral current is
3
since ( ) / 3
If the system is balanced, then 0 and
0
This is true for
n a b c
a b c
a b c
I I I I I
I I I I
I I I
I
any 3-ph system with no neutral path,
eg 3ph -connnected or 3 wire Y-connected systems.
Symmetrical Components
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Symmetrical Component Example 1
Symmetrical Components
20
1 2s
2
10 0
Consider 10 Then
10
1 1 1 10 01
1 10 10 03
10 01
There is only positive sequence component for since
a
b
c
a b
I
I
I
I I
I
I A I
I
and are balanced positive sequencec I
Symmetrical Components
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Symmetrical Component Example 1
Symmetrical Components
21
s
10 0 0
If 10 then 0
10 10 0There is only negative sequence component for since
and are balanced negative sequencea b c I I I
I I
I
Symmetrical Components
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Symmetrical Component Example 2
Symmetrical Components
22
1 2s
2
0
Let unsymmetrical 3ph voltages
Then
1 1 1 0 01
13
6.121There are 3 sequence comp
a
b
c
V
V
V
V
V A V
onents for since
, and are unsymmetrical 3ph voltagesa b cV V V
V
Symmetrical Components
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Symmetrical Component Example 3
Symmetrical Components
23
0
1
2
2
2
10 0Let 10 with 3 sequence components
Then1 1 1 10 0
1 10
1
contains unbalanced 3ph curren
s
s
I I
I
I
I AI
ts
Symmetrical Components
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Symmetrical Component Example 4
Symmetrical Components
24
Symmetrical Components
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Symmetrical Component Example 4
y p
25
Symmetrical Components
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Symmetrical Component Example 4
y p
26
Symmetrical Components
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Outline
• Concept and definition of symmetrical components
• Sequence networks of impedance loads
• Sequence networks of series impedances
• Sequence networks of transmission and distribution lines
• Sequence networks of rotating machines
• Sequence networks of transformers
• Power in sequence networks
•Main reference: Ch 8 Textbook
y p
27
Symmetrical Components
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Sequence Networks of Impedance Loads
• A note about symbols
y p
28
Symmetrical Components
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g − ground V ng = I n Z n
N
Sequence Networks of Impedance Loads
• Consider the following Y-connected load:
29
( )
Similarly
( )
( )
n a b c
ag a y n n
ag Y n a n b n c
bg n a Y n b n c
cg n a n b Y n c
I I I I
V I Z I Z
V Z Z I Z I Z I
V Z I Z Z I Z I
V Z I Z I Z Z I
Symmetrical Components
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Sequence Networks of Impedance Loads
30
In matrix form
Write and in sequence component form
ag y n n n a
bg n y n n b
ccg n n y n
s
s
V Z Z Z Z I
V Z Z Z Z I
I V Z Z Z Z
IV Z
V Z I
V I
V A V
I A I
Symmetrical Components
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Sequence Networks of Impedance Loads
31
1
1
Then can be written as
3 0 0
0 0
0 0
sequence component impedance matrix
s s
s s s s
y n
s y
y
s
Z Z
Z
Z
V ZI
AV ZAI
V A ZAI Z I
Z A ZA
Z
Symmetrical Components
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Sequence Networks of Impedance Loads
32
1 2 2
2 2
2 2
2 2
2
1 1 1 1 1 111 1
31 1
31 1 111 3
31 3
3 0 00 0 , using 1+
0 0
y n n n
n y n n
n n y n
y n y y
y n y y
y n y y
y n
y
y
Z Z Z Z Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z Z
Z Z Z
Z
A ZA
30, 1
Symmetrical Components
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Sequence Networks of Impedance Loads
33
0 0
1 1
2 2
0 0
1 1
2 2
3 0 00 0
0 0
Systems are decoupled( 3 )
y n
y
y
y n
y
y
Z Z V I V Z I
V I Z
V Z Z I
V Z I
V Z I
• With balanced impedances,
the three phase Y-connected
load can be represented by
three completely decoupled
zero, positive and negativesequence networks
Symmetrical Components
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Sequence Networks of Impedance Loads
34
I 1
I 2
I 0
Z y
Z y
Z y 3Z n
Z 1 = Z y Z 2 = Z y Z 0 = Z y + 3 Z n
• If neutral is not grounded
via Z n then the zero
sequence is open circuit
Symmetrical Components
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Sequence Networks of Impedance Loads
35
Balanced -connected impedance load and its Y-equivalent
Z /3 Z
/3
Z /3
Z Z
Z
•
Balanced
-connected impedance loads
Symmetrical Components
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Sequence Networks of Impedance Loads
36
Sequence networks for the Y-equivalent of a balanced load
• Since the -connected load
does not have a neutralconnection, the equivalent
Y-connected load has an
open circuited neutral
I 1
I 2
I 0
Z /3
Z /3
Z /3
Symmetrical Components
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Sequence Networks Example
37
Sequence networks: balanced Y and balanced loads
A balanced Y-load is in parallel with a balanced -connected
capacitor bank. The Y-load has an impedance Z y = (3 + j4)
per phase, and its neutral is grounded through an inductive
reactance X n= 2 . The capacitor bank has a reactance X c =
30 per phase. Draw the sequence networks for this loadand calculate the load sequence impedance.
Solution
Symmetrical Components
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Sequence Networks Example
38
Equivalent circuit and sequence networks
//
//
//
Symmetrical Components
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Sequence Networks Example
39
Symmetrical Components
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Sequence Networks Example
40
Symmetrical Components
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Outline
• Concept and definition of symmetrical components
• Sequence networks of impedance loads
• Sequence networks of series impedances
• Sequence networks of transmission and distribution lines
• Sequence networks of rotating machines
• Sequence networks of transformers
• Power in sequence networks
•Main reference: Ch 8 Textbook
41
Symmetrical Components
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Sequence Networks of Series Impedance
Elements
42
Z a, Z b and Z c – self-impedances for each phase
Z ab, Z bc and Z ac – mutual-impedances between phases
n
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Sequence Networks of Series Impedance
Elements
43
The voltage drops across the series-phase impedances [betweenthe buses (a,b,c) and (a′,b′,c′ )] are given by
Transforming to symmetrical components gives
Or equivalently
'
'
'
an a n a ab ac a
bn b n ab b bc b
cn c n ac bc c c
V V Z Z Z I
V V Z Z Z I
V V Z Z Z I
0 0' 0
1 1' 1
2 2' 2
a ab ac
ab b bc
ac bc c
V V Z Z Z I
A V A V Z Z Z A I V V Z Z Z I
Symmetrical Components
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Sequence Networks of Series Impedance
Elements
44
'
0 0' 01
1 1' 1
2 2 ' 2
'
1
2
2
1 1 11
13
Sequence Imped
1
ance:
s s s s
a ab ac
ab b bc
ac bc c
s s s s
a ab ac
s ab b bc
ac bc c
V V Z Z Z I V V A Z Z Z A I
V V Z Z Z I
Z Z Z
A Z Z Z A
Z Z Z
V V IZ
V V Z I
Z
2
2
1 1 1
1
1
a ab ac
ab b bc
ac bc c
Z Z Z
Z Z Z
Z Z Z
Symmetrical Components
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Sequence Networks of Series Impedance
Elements
45
2 2
2 2
0
1
2
1 1 1 1 1 11
1 13
1 1
0 0 2 0 0
0 0 0 0
0 0 0 0
a ab ac
s ab b bc
ac bc c
a ab
a ab
a ab
Z Z Z
Z Z Z
Z Z Z
Z Z Z
Z Z Z
Z Z Z
Z
For symmetrical series impedances
,a b c ab ac bc Z Z Z Z Z Z
Sequence Impedance Matrix is thus given by
Symmetrical Components
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Sequence Networks of Series Impedance
Elements
46
0 0' 0 0 0
1 1' 1 1 1
2 2 ' 2 2 2
( 2 )
( )
( )
a ab
a ab
a ab
V V Z I Z Z I
V V Z I Z Z I
V V Z I Z Z I
• The three sequence
circuits are decoupled
Symmetrical Components
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Outline
• Concept and definition of symmetrical components
• Sequence networks of impedance loads
• Sequence networks of series impedances
• Sequence networks of transmission and distribution lines
• Sequence networks of rotating machines
• Sequence networks of transformers
• Power in sequence networks
•Main reference: Ch 8 Textbook
47
Symmetrical Components
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Ground Current Paths
48
• The zero sequence currents of the phases of overhead transmission lines
cause a field that links the earth and the associated earth currents
• A “remote Equipotential Ground” conductor is derived that has an
equivalent effect to the ground current
Eddy currents reduceground penetration
H field due to line current
IL
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Symmetrical Component of Distribution Lines
49
• Distribution lines
usually do not have
earth wire/s
• The ground currents
only flow in theground
• Zero sequence
causes earth
currents
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Symmetrical Component of Distribution Lines
50
• A current Ig occurs in the ground due to the zero
sequence currents.
• The magnetic flux associated with the ground current
interacts with the magnetic flux associated with the phase
currents.
Symmetrical Components
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Symmetrical Component of Transmission Lines
51
• Transmission lines
usually have overhead
earth wires.
• These earth wires
shield the line fromlightening strikes.
• They act as a parallel
path to the earth for
“Ground-currents”
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Symmetrical Component of Transmission Lines
52
Symmetrical Components
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Symmetrical Component of Transmission Lines
53
• Ground-current flows in the ground and earth wires• The ground-current defines the impedance of the ground
path
Z g = R g + j X g
• Z g modifies the self and mutual impedances of the series
element such that
'
'
a a a g
ab ab ab g
Z Z Z Z
Z Z Z Z
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Symmetrical Component of Transmission Lines
54
Therefore the sequence impedance matrix
0
1
2
0 0 2 0 0
0 0 0 0
0 0 0 0
a ab
s a ab
a ab
Z Z Z
Z Z Z
Z Z Z
Z
becomes
' ' '0
' ' ' '
1
' ' '2
0 0 2 0 0
0 0 0 00 0 0 0
a ab
s a ab
a ab
Z Z Z
Z Z Z Z Z Z
Z
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Symmetrical Component of Transmission Lines
55
Practical implication' '
0
' '1
2 2 2a ab a ab g
a ab a ab g
Z Z Z Z Z Z
Z Z Z Z Z Z
Generally Z 0 > Z 1
Examples
66kV line – no earth wires 100km
Z 1= 0.85 + j1.1, Z 0 = 1.25 + j4.3
500kV line 2 earth wires 150km
Z 1= 0.001 + j0.016, Z 0 = 0.008 + j0.038
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Outline
• Concept and definition of symmetrical components
• Sequence networks of impedance loads
• Sequence networks of series impedances
• Sequence networks of transmission and distribution lines
• Sequence networks of rotating machines
• Sequence networks of transformers
• Power in sequence networks
•
Main reference: Ch 8 Textbook
56
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Sequence Networks of Generators
57
Y-connected synchronous
generatorSequence networks of Y-connected
synchronous generator
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Sequence Networks of Generators
58
Positive Sequence
• V1 sequence rotates in the same direction as the
generator rotor
• Generates the rotating magnetic field
• Only sequence with a source Eg1
Negative Sequence• V2 sequence rotates in the opposite direction to the
generator rotor
• Induces negative sequence rotor current with twice
frequency (eg 250Hz = 100Hz) that reduces flux penetration of rotor, resulting in reduced reactance, ie
• Zg2 < Zg1
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Sequence Networks of Synchronous Motors
59
• Currents flow into the
networks – absorbing power
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Sequence Networks of Induction Motors
60
• Positive network has
no voltage source sinceinduction motor does
not have back emf
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Sequence Networks of Rotating Machines Example 1
61
Example 8.5
A balanced, positive sequence, Y-connected generator withinternal voltage E ab = 4800° V is applied to a balanced -
load with Z =3040° . The line impedance between the
source and the load is Z L=185° for each phase and the
transmission line mutual coupling is ignored. Assume that thegenerator is grounded via an impedance Z n= j10 and that the
generator sequence impedances are Z g 0= j1 , Z g 1= j15 and
Z g 2= j3 .
• Draw the sequence networks
• Calculate the sequence component of the line current
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Sequence Networks of Rotating Machines Example 1
62
0 1 2
480
480 0 303
1 , 1 85 , 30 40
1 , 15 , 3
ab a b a
n L
g g g
E E E E
Z j Z Z
Z j Z j Z j
Solution
Symmetrical Components
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Sequence Networks of Rotating Machines Example 1
63
0
21
22
1
2 0
Sequence component voltages of the generator
1 1 1
11
31
The source is positive-sequence and balanced with phase voltage
48030
3
0
g a
g b
cn g
a
g a
g g
E E
E E
E E
E
E E
E E
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Sequence Networks of Rotating Machines Example 1
64
From
Symmetrical Components
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Sequence Networks of Rotating Machines Example 1
65
and
we get
Z ∆ Z ∆
Z ∆
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Sequence Networks of Rotating Machines Example 1
66
I 1
I 2
I 0
Z /3
Z /3
Z /3
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Sequence Networks of Rotating Machines Example 1
67
0 2
1
1
1 1
0
/ 3
11.68 100.94
g
g L
I I
E I
Z Z Z
1
48030
3 g E
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Sequence Networks of Rotating Machines Example 2
68
Example 8.6
If a solidly grounded Y-connected voltage source with the
unbalanced terminal voltages given below is applied to the
balanced load in the previous example, calculate the phase
currents.
0ag
bg
cg
V
V
V
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Sequence Networks of Rotating Machines Example 2
69
Solution The sequence components of the source voltages are
02
1
22
1 1 11
13
1
1 1 1 01
1 120 1203
1 120 120
62.11
9.218
ag
bg
cg
V V
V V
V V
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Sequence Networks of Rotating Machines Example 2
70
0
11
1
22
1
0 02
1 1
2
2 2
0 an open circuit
/ 3
25.82 45.55
/ 30.86 172.82
1 1 1
1
146.76
L
L
a
b
c
I
V I
Z Z
V I
Z Z
I I I
I I I
I I I
A
2=
9.218∠− 143.41°
0=
15.2∠62.11°
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Outline
• Concept and definition of symmetrical components
• Sequence networks of impedance loads
• Sequence networks of series impedances
• Sequence networks of transmission and distribution lines
• Sequence networks of rotating machines
• Sequence networks of transformers
• Power in sequence networks
•Main reference: Ch 8 Textbook
71
S N t k f T f
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Sequence Networks of Transformers
72
Single-line
diagram
Symbols
Sequence Networks of Transformers
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Sequence Networks of Transformers
• Transformers are a series element
• Positive and negative sequences are the phase impedancesrepresenting flux leakage effects and the resistance of the
windings
73
• The excitation branch
of the transformermodel is neglected
when considering
sequence models for
the purposes of faultcalculations
Zero Sequence Networks of Transformers
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Zero Sequence Networks of Transformers
•Zero sequence equivalent circuit is determined by thewinding arrangement, ie Y or and earthing of Y
connected windings
•Zero sequence currents, I a0 , I b0 and I c0 are in phaseThus, a delta winding acts as a short circuit for zero
sequence currents
74
Zero Sequence Networks of Transformers
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Zero Sequence Networks of Transformers
•There can only be a current flowing in a particular phaseof a primary winding, if it is possible for the equivalent
current to flow in the corresponding winding of the
transformer secondary
•
Zero sequences currents cannot flow in the secondary because no neutral and hence no primary current
• Zero sequence open circuit on secondary
75
Zero Sequence Networks of Transformers
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Zero Sequence Networks of Transformers
• Zero sequences currents can flow in the secondary becauseof earthed neutral
• Induces primary current which flows because of earthed
neutral
• Zero sequence series circuit primary to secondary
(Must include neutral grounding Z if neutral grounded
through impedance)
76
Zero Sequence Networks of Transformers
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Zero Sequence Networks of Transformers
• Zero sequences currents can flow through primary and
within the secondary because of delta
• But no flows in secondary connection lines because current
confined to delta, ie open circuit on secondary side
• If neutral grounded through impedance Z n, then a 3 Z n is in
series connection with the Z 0 of the transformer
77
Zero Sequence Networks of Transformers
Symmetrical Components
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Zero Sequence Networks of Transformers
• Zero sequences currents cannot flow in the primary since
unearthed neutral
No primary current and no secondary current
Open circuit on secondary side
78
Zero Sequence Networks of Transformers
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Zero Sequence Networks of Transformers
79
Dual to Y-∆ cases : circuits flipped
Zero Sequence Networks of Transformers
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Zero Sequence Networks of Transformers
• Zero sequences currents flow in the secondary because ofdelta
• Induces currents in primary delta
• But open circuit on primary and secondary connection
lines because current in connection lines is confined todelta
80
Transformer Sequence DiagramsSymmetrical Components
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Transformer Sequence Diagrams
81
Transformer Phase Shifts
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Transformer Phase Shifts
•
Previous models ignore the phase shift in Y-
and
-Y• Include phase shift in Y- and -Y based on the
convention:
o The positive sequence voltages and currents for the high
voltage side of the Y-
transformer lead thecorresponding quantities on the low voltage side of the
transformer by 30°
o Hence the negative sequence voltages and currents for
the high voltage side of the Y- transformer must lag
the corresponding quantities on the low voltage side by
30°
82
Transformer Phase Shifts
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Transformer Phase Shifts
• Note that although in practice the zero sequence
impedance of a transformer may differ slightly to the
positive and negative sequence impedances, often the same
numeric value is used for all three impedances
83
Symmetrical Components
Sequence Networks of Transformers Example
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Example 8.7
84
Sequence Networks of Transformers Example
02
1
22
1 1 1 62.1111
39.2181
ag
bg
cg
V V V V
V V
From Example 8.6 the sequence voltages are
Symmetrical Components
Sequence Networks of Transformers Example
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85
Sequence Networks of Transformers Example
Fig on p70.
Symmetrical Components
Sequence Networks of Transformers Example
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86
Sequence Networks of Transformers Example
Symmetrical Components
Sequence Networks of Transformers Example
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87
Sequence Networks of Transformers Example
Symmetrical Components
Sequence Networks of Transformers Example
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88
Sequence Networks of Transformers Example
Symmetrical Components
Sequence Networks of Transformers Example
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89
Sequence Networks of Transformers Example
O tli
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Outline
• Concept and definition of symmetrical components• Sequence networks of impedance loads
• Sequence networks of series impedances
• Sequence networks of transmission and distribution lines
• Sequence networks of rotating machines
• Sequence networks of transformers
• Power in sequence networks
•
Main reference: Ch 8 Textbook
90
Symmetrical Components
Power in Sequence Networks
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91
Power in Sequence Networks
Total power delivered to the 3-phase load
Symmetrical Components
Power in Sequence Networks
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92
Power in Sequence Networks
*
*
* * * * *
*
* * *
Total power is given by
Using and gives
( ) ( )
T p
p
p
a
T
p ag a bg b cg c ag bg cg b p p
V c
I
p s s
T T
p s s s s
S
I
S V I V I V I V V V I V I
I
V V I I
S V I V I
A A
A A AA
Symmetrical Components
Power in Sequence Networks
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93
Power in Sequence Networks
*
* 2 2
2 2
2 2
2 2
*
0
* * * * *
0 1 2 1 0 0 1 1 2 2
*
2
1 1 1 1 1 1
1 1
1 1
1 1 1 1 1 1 1 0 0
1 1 3 0 1 0
1 1 0 0 1
3 3 3( ) 3T
p s s s
I
S V I V V V I V I V I V I S
I
AA
Symmetrical Components
Power in Sequence Networks Example
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94
Power in Sequence Networks Example
Example 8.9
Symmetrical Components
Power in Sequence Networks Example