Review of Three-Phase Circuit Analysis
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
Three-phase system
vbnvan vcn
Time
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.
Y-Connection-Source
a
b
c
a' b'
c' n Vaa
Vbb
Vcc a
a'
Vaa
b
b'
c
Vbb Vcc
c'
n
nVVVan 0
Van
Vbn
Vcn
Reference
Vcn Vbn Vann
cb
a
Phase voltage
a
c
b
VVVcn 120VVVbn 120
Van Vbn Vcn
GeneratorTransmission
Line
VVVVV phcnbnan
90V3120V120VVVV cnbnbc
Line-to-line voltage
30V3120V0VVVV bnanab
150V30V120VVVV ancnca
Vca Vbc
Vab
n
cb
a
Van Vbn Vcn
303
23
21
)120sin120(cos1
1200
ph
phph
ph
bnanab
V
jVV
jV
VVVVV Reference
VabVca
nVbn
Vcn
Van
Vbc
030
-Vbn
The other line to line voltages
oabca
oabbc
VV
VV
120
120
30V3120V0VVVV bnanab Line-to-line voltage is greater than phase voltage by
3 Line-to-line voltage leads phase voltage by
030
Main Conclusions
ExampleLet for a balanced three phase system
VVan 0240
Calculate the line-to-line voltages
oooabo
anab
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
Y-Connected System
Source LoadTransmission
Line
a
c b
n
+
Van Z
Z Z
Ia
Ib
Ic
VbnVcn
a
bcn
Ia Ia
Ic IcIb
Ib
Line current
Phase current
+
++ + +
Three Phase System
120
120
0
VVVVVV
cn
bn
an
)120(120
)120(120
0
o
c
cnc
o
b
bnb
o
a
ana
IZV
ZVI
IZ
VZVI
IZV
ZVI
Y- Connection
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
nVan
Vbn
Vcn
Ia
Ic
Ib
120
120
0
VVVVVV
cn
bn
an
)120(II
)120(II
II
oc
ob
oa
Reference
Neutral Current of Balanced Load
ZZZZ cba For balanced system
01201200 oocban IIIIIII
Ia
Ib
Ic
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
Ia
Ic
a
bcbcIb
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
AjZ
VI ooo
ana 13.53241.535
0120430120
87.6625)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
Delta () Connection: Loada
bc
Ia
Ib
Ic
IabIca
Ibc
+_
+_+
_
Ibc Iab
nVab
Vbc
Vca
Refere
Ica
Z
VIZ
VIZ
VI cacabcbcabab ,,
Ia
caaba III
IbcIab
nVab Refere
Ica
300
)30(I3I
30I3III
oaba
oabcaaba
- Ica
Line current lags phase current by
030
Main Conclusions for Delta-Connected Load
Line-to-line voltage is equal to phase voltage
)30(I3I oaba
3 Line current is greater than phase current by
Example
bc
Ia
Ic
a
bc
Ib
Calculate the phase currents of the load
VVan 40120 VZ 3010
Ibc
IabIca
abc
Ia
Ic
VanbcIb
Z
AZ
VI ooo
abab 757.41375
301203
oooabo
anab
V
VV
307.4153002403
303
Power of 3-phase circuits
Iphase
VphaseFor Single phase
)(sin
)(cos
phasephase
phasephase
IVQIVP
For 3-phase
)(sin3
)(cos3
phasephase
phasephase
IVQIVP
IMPORTANT
is the angle between phase voltage and phase current.
Use voltage as a reference
Real Power in Y Circuit
aphase
anphase
II
VV
aline
anabline
IIVVV
3
)(cos3)(cos3 aanphasephase IVIVP )(cos3)(cos3 linelineaab IVIVP
a
bc
Ia
Ib
Ic
Van
Vcn Vbn
Reactive Power in Y Circuit
aphase
aphase
IIVV
aline
aabline
IIVVV
3
)(sin3)(sin3 aaphasephase IVIVQ )(sin3)(sin3 linelineaab IVIVQ
a
bc
Ia
Ib
Ic
Va
Vc Vb
Real Power in Delta Circuita
bc
Ia
Ib
Ic
IabIca
Ibc
+_
+_+
_
abphase
abphase
IIVV
abaline
abline
III
VV
3
)(cos3)(cos3 ababphasephase IVIVP )(cos3)(cos3 linelineaab IVIVP
Reactive Power in Delta Circuita
bc
Ia
Ib
Ic
IabIca
Ibc
+_
+_+
_
abphase
abphase
IIVV
abaline
abline
III
VV
3
)(sin3)(sin3 ababphasephase IVIVQ )(sin3)(sin3 linelineaab IVIVQ
Review of Three-Phase Circuit Analysis Slide Number 2Three-phase systemWhy do we use 3-phase systems?Y-Connection-SourcePhase voltageSlide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Main ConclusionsExampleY-Connected SystemSlide Number 15Y-ConnectionSlide Number 17Neutral Current of Balanced LoadSlide Number 19Slide Number 20Slide Number 21Delta () Connection: LoadSlide Number 23Slide Number 24ExampleSlide Number 26Power of 3-phase circuitsSlide Number 28Slide Number 29Slide Number 30Slide Number 31
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