chapter 12 three-phase circuit analysis. i a i b i c n s n s n s n s n s n s vava v b vcvc n s n s a...
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![Page 1: Chapter 12 Three-Phase Circuit Analysis. I a I b I c N S N S N S N S N S N S VaVa V b VcVc N S N S A three phase generator has three coils each represent](https://reader035.vdocuments.site/reader035/viewer/2022081421/56649d895503460f94a6f355/html5/thumbnails/1.jpg)
Chapter 12Three-Phase Circuit Analysis
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Ia
Ib
Ic
N S
N
S
N
S
N
S
NS
N
S
Va
Vb
Vc
N
S
N
S
A three phase generator has three coils each represent a phase
Vc
Vb
Va
2
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Three-phase system
3
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Why do we use 3-phase systems?• Three-phase system produces rotating magnetic
field.– Three-phase motors can start without the need for
extra equipment.• For the same physical size, a three-phase
generator produces more power than a single phase generator.
• Three-phase lines transmit more power.• Three-phase lines are more reliable.
– In distribution circuit, you can operate the system with one missing phase.
4
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Y-Connection-Source
a
b
c
a' b'
c'
n Vaa’
Vbb’
Vcc’ a
a'
Vaa’
b
b'
c
Vbb’ Vcc’
c'
n
5
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n
VVVan 0
Van
Vbn
Vcn
Reference
Vcn Vbn Vann
c
b
a
Phase voltage
a
c
b
VVVcn 120
VVVbn 120
Van Vbn Vcn
GeneratorTransmission
Line
VVVVV phcnbnan
6
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7
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9
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90V3120V120VVVV cnbnbc
Line-to-line voltage
30V3120V0VVVV bnanab
150V30V120VVVV ancnca
Vca Vbc
Vab
n
c
b
a
Van VbnVcn
10
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303
2
3
2
1
)120sin120(cos1
1200
ph
phph
ph
bnanab
V
jVV
jV
VVVVV Reference
VabVca
n
Vbn
Vcn
Van
Vbc
030
-Vbn
The other line to line voltages
oabca
oabbc
VV
VV
120
120
11
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30V3120V0VVVV bnanab
• Line-to-line voltage is greater than phase voltage by
3
• Line-to-line voltage leads phase voltage by
030
Main Conclusions
12
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Example
Let for a balanced three phase system
VVan 0240
Calculate the line-to-line voltages
oooab
oanab
V
VV
307.4153002403
303
Reference030
van
vab
oobc
oabbc
V
VV
907.415120307.415
120
ooca
abca
V
VV
1507.415120307.415
120
The other voltages can be computed by the balanced system relationship
13
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Y-Connected System
Source LoadTransmission
Line
a
c b
n
+
Van Z
Z Z
Ia
Ib
Ic
Vbn Vcn
a
b c
n
Ia Ia
Ic Ic Ib
Ib
Line current
Phase current
+
++ + +
14
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Three Phase System
15
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120
120
0
VV
VV
VV
cn
bn
an
)120(120
)120(120
0
o
c
cnc
o
b
bnb
o
a
ana
IZ
V
Z
VI
IZ
V
Z
VI
IZ
V
Z
VI
Y-Connecti
on
ZZZZ cba For balanced system
a
c b
n
+
Van Z
Z Z
Ia
In
Ib
Ic
Vbn Vcn
a
b c
n
Ia Ia
Ic Ic Ib
Ib
16
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nV
an
Vbn
Vcn
Ia
Ic
Ib
120
120
0
VV
VV
VV
cn
bn
an
)120(II
)120(II
II
oc
ob
oa
Reference
17
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Neutral Current of Balanced Load
ZZZZ cba For balanced system
01201200 oocban IIIIIII
Ia
Ib
Ic
18
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30V3120V0VVVV bnanab
• Line-to-line voltage is greater than phase voltage by
3
• Line-to-line voltage leads phase voltage by 030
Main Conclusions for Y-Connected Load
• Line current equals phase current
19
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Ia
Ic
a
bc
bcIb
Example: For the three phase system shown , findThe load currents of each phaseThe neutral currentThe magnitude of the line to line voltage
VVan 0120 43 jZ
20
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AjZ
VI
o
ooan
a 13.53241.535
0120
43
0120
87.6624)13.53120(24)120(
13.17324)13.53120(24)120(
13.5324
ooc
oob
ooa
II
II
II
The other currents can be computed by the balanced system relationship
087.662413.1732413.5324 oocban IIII
The load currents of each phase
The neutral current
The magnitude of the line to line voltage
VVV phL 20812033 21
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Delta () Connection: Loada
bc
Ia
Ib
Ic
IabIca
Ibc
+
_
+_
+
_
Ibc Iab
nV
ab
Vbc
Vca
Reference
Ica
Z
VI
Z
VI
Z
VI ca
cabc
bcab
ab ,,
22
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Ia
caaba III
IbcIab
nV
ab Reference
Ica
300
)30(I3I
30I3III
oaba
oabcaaba
- Ica
23
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Main Conclusions for Delta-Connected Load
• Line-to-line voltage of the source is equal to phase voltage (across load).
• The line current (coming from the source) is greater than phase current (of the load) by
•Line current lags phase current by
24
)30(3 oaba II
3
030
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Example
bc
Ia
Ic
a
bc
Ib
Calculate the phase currents of the load
VVan 0120
VZ 3010
Ibc
IabIca
25
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a
bc
Ia
Ic
Van
bcIb
Z
AZ
VI o
o
oab
ab 079.203010
301203
oooab
oanab
V
VV
309.2073001203
303
26
ooooaba II 3036301079.203303
![Page 27: Chapter 12 Three-Phase Circuit Analysis. I a I b I c N S N S N S N S N S N S VaVa V b VcVc N S N S A three phase generator has three coils each represent](https://reader035.vdocuments.site/reader035/viewer/2022081421/56649d895503460f94a6f355/html5/thumbnails/27.jpg)
Power of 3-phase circuits
Iphase
VphaseFor Single phase
)(sin
)(cos
phasephase
phasephase
IVQ
IVP
For 3-phase
)(sin3
)(cos3
phasephase
phasephase
IVQ
IVP
IMPORTANT
is the angle between phase voltage and phase current.
Use voltage as a reference 27
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Real Power in Y Circuit
aphase
anphase
II
VV
aline
anabline
II
VVV
3
)(cos3)(cos3 aanphasephase IVIVP
)(cos3)(cos3 linelineaab IVIVP
a
bc
Ia
Ib
Ic
Van
Vcn Vbn
28
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Reactive Power in Y Circuit
aphase
aphase
II
VV
aline
aabline
II
VVV
3
)(sin3)(sin3 aaphasephase IVIVQ
)(sin3)(sin3 linelineaab IVIVQ
a
bc
Ia
Ib
Ic
Va
Vc Vb
29
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Real Power in Delta Circuit
a
bc
Ia
Ib
Ic
IabIca
Ibc
+
_
+_
+
_
abphase
abphase
II
VV
abaline
abline
III
VV
3
)(cos3)(cos3 ababphasephase IVIVP
)(cos3)(cos3 linelineaab IVIVP 30
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Reactive Power in Delta Circuit
a
bc
Ia
Ib
Ic
IabIca
Ibc
+
_
+_
+
_
abphase
abphase
II
VV
abaline
abline
III
VV
3
)(sin3)(sin3 ababphasephase IVIVQ
)(sin3)(sin3 linelineaab IVIVQ 31
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Three Phase Power Measurement Two-meter method for measuring three-phase power