§ 12-3 The delta( ) connection?
A --connection three-phase sources?
a l sm i nt e rl i n ec,b,a ??
?0??? ?? papan VVV
?1 2 0???? ?? pbpbn VVV
?1 2 0???? ?? pcpcn VVV
pL VV ?? ?
0??? ??? cpbpap VVV
apV?
bpV?
cpV?
n
apV?
bpV?
cpV?
n
bpap VV ?? ?
cpcpbpap VVVV ???? ???? 2
?c
?
?
a
?
?
?
?
b
?
?
a
b
c
apV
?
bpV
?
cpV
?
c ?
a
?
?
? b
bpV
?
?
?
cpV
?
apV
?
Y-- connection?
cnbnanp
cabcabL
VVVV
VVVV
???
???
pL VV 3?
?? 303303 ???? ?? panab VVV ?903 ???? pbc VV ?1 5 03 ???? pca VV
-- connection load:?
p
CA
CA
p
BC
BC
p
AB
AB Z
VI
Z
VI
Z
VI
?
?
?
?
?
?
???,,
The phase currents
are also balance.
CABCAB III
???,,
CABCABpCABCAB IIIIa n dIII ??????
??? 0
Y--source:
)ZZZZ( pCABCAB ???
?
?
?a b
c
n
? bnV?
?
? ?
?
? ?
?
?
?A B
C
pZp
Z
pZ
anV
?
cnV
?
The line currents:
CAABaA III ??? ??
ABBCbB III ??? ??
BCCAcC III
??? ??
LcCbBaA
cCbBaA
IIIIan d
III
???
???
???
0
pL II 3?aAI
?
bBI?
bnV?
cnV?
bcV?
caV?
abV?
anV?
?30 ABI?
BCI?
CAI?
cCI?
?? ??? 303 ABI
?? ??? 303 BCI
?? ??? 303 CAI
?
?
?a b
c
n
? bnV?
?
? ?
?
? ?
?
?
?A B
C
pZp
Z
pZ
anV
?
cnV
?
Example,We are to determine the magnitude of the line
current in VL=300V three-phase system which supplies
12000W to a -connected load at a lagging PF of 0.8.?
Solution:
8.0c o s,40 003/12 000,30 0 ???? ?WPVV ABL
AI p 7.168.0300 4000 ????
AII pL 9.283 ???
??????? ?? 8.104.149.36188.0c o s7.16300 1 jZ p
The three-phase power
We assume a Y-connected load with a power-factor angle,?
?? co s31co s LLppp IVIVP ??
The total power
?? c o s3c o s3133 LLLLp IVIVPP ?????
The power delivered to each phase of a - connection load.?
??? c o s31c o s31c o s LLLLppp IVIVIVP ???
The total power
?? c o s3c o s3133 LLLLp IVIVPP ?????
?? co s3co s3 VIIVP LL ??
Example,If VL=380V,Zp=4.5+j6Ω,ZL=1.5+j2 Ω.
Find
?
?
?a b
c
n?
bnV?
?
? ?
?
? ?
?
?
?A B
C
pZ
pZ
pZ
anV?
cnV? LZ
LZ
LZ;,,,,)1( CABCABcCbBaA IIIan dIII ??????,P)( load2
Solution:
VVl et)( an ?? 0220033801 ?????
AZZ VI
pL
an
aA ?1.53443/ ?????
?
?
???anV
???bnV
???cnV
LZ
LZ
LZ
3/pZ
Nn
a A
3/pZ
3/pZ
??? anV
N
a A
LZ 3/pZ
AII aAAB ?? 1.234.25303 ?????
?
?
W..
RIP)( ppl oa d
8 7 1 0544253
32
2
2
????
???
The three-phase circuits analysis
1,A balanced three-phase circuit
(1) The balanced three-phase system => Y-Y;
(2) Draw a phase circuit (A phase),then find voltages
and currents of a phase(A phase);
(3) Direct write other two phase(B,C) voltages and
currents from the balanced relationship;
(4) Find voltages and currents of the original circuit
from the Y- transformation.?
Example,
a
b
c
LZ 1Z
2Z
A
B
C
If,ZL=j5,Z1=50,
Z2=90+j120,VL=380V,
(+)sequence.
Find the phase voltages
and currents of two loads.
a
b
c
LZ 1Z
3/2Z
A
B
C
a
LZ 1Z
3/2Z
A
n
AI
?
1AI
?
2AI
?
Solution:
a
LZ 1Z
3/2Z
A
n
AI
?
1AI
?
2AI
?
Solution:
VVL et an ?? 022003380 ????? A
ZZ
ZZZ
VI
L
an
A
?352.7
3/
3/
21
21
???
?
???
?
?
A../ZZ /ZII AA ??? ?????? 5804433
21
21 A
ZZ
ZII
AA ?5.6104.43/
21
12 ???
???
??
AII AAB ?? 5.3133.2303 2 ?????
?
?
VVVV pLp 3503 ????
V.IZV Ap 2 0 20445011 ???? ?
a
b
c
LZ 1Z
2Z
A
B
C
DP,If ZL=j0.6Ω,Z2=4.8+j3.6 Ω,Z1{VL1=300V,P1=12000W,
PF1=0.8(lagging)},Find VSL.
a
LZ 3/1Z
2Z
A
n
AI
?
1AI
?
2AI
?
a
b
c
LZ
1Z
A
B
C
2Z
Solution:
AIIVP ppL 7.168.03003 1 2 0 0 0c o s3 11111 ???????
187.16300
1
1
1 ???
p
L
I
VZ 8.104.149.36188.0c o s18 11 jZ ?????? ? ?
VVL et An ?033 0 0 ??? ?9.3635012 ????
??
AA II ?9.363
100
21 ?????
???
AAA III
?13.8
3
340 ???? ???
AnALan VIZV
VVV ansL 3403 ??
DP,Find the line current,the total power delivered to the load
and the total power-loss in the lines for the circuit,if
.5.0,35.7,0180 ???????? Lpab ZjZVV ?
a
n
5.0
15.2 j?
anI
?
Solution:
VVV aban ?? 3031 8 0303 ??????
?
?
A..j.,VI anan ?44893215250 ???????
?
?
AIAI pL 19,9.32 ??
WP l o a d 8 1 2 25.7193 2 ????
WP l i n e 1 6 2 45.09.323 2 ????
?
?
?a b
c
n
bnV?
?
? ?
?
? ?
?
A
B
C
pZ
pZ
pZ
anV?
cnV?
LZ
LZ
LZ
2,An unbalanced three-phase circuit
?
?
?a b
c
n?
anV?
?
? ?
??
?
?
?
?A B
C
?
CZ
BZ
AZ
N
bnV?
cnV?
NZ
cnbnan
cnbnan
VVVan d
0VVV
??
???
???
CBA ZZZ ??
0
1111
?
???
??
?
???
?
NCBA
C
cn
B
bn
A
an
Nn
ZZZZ
Z
V
Z
V
Z
V
V
A
AN
ANNnanAN Z
VIVVV
?
????
????
C
CN
CNNncnCN Z
VIVVV
?
????
????
B
BN
BNNnbnBN Z
VIVVV
?
????
????
bnV
?
cnV
?
anV
?
BNV
?
ANV
?
CNV
?
NnV
?
Nn
N
Nn
Nn
VIa n d
?
?
?
00 ?? ? NnN VZat
v ol t ag e sba l an c e dVVV CNBNAN ????,,
The phase sequence indicator(12-4)
N
C
2R
1R
A
ph a s eACRCRR ????? ?/121
pp VjVj
j )6.02.0(
12
11 ???
?
???
2
12 012 0
?
??????
j
VVjV ppp ??
RR
Cj
R
V
R
V
VCj
V
cnbn
an
Nn 11
??
??
?
??
?
?
?
?
)( l i g h tp h a s eBVV CNBN ???
bnV
?
cnV
?
anV
?
BNV
?
ANV
?
CNV
?
NnV
?
N
n
The phase sequence indicator(12-4)
N
C
2R
1R
A
ph a s eACRCRR ????? ?/121
pV).j.( 6020 ??
?? NnV
?? 5101516020120,V.V).j.(VVVV pppNnbnBN ??????????? ???
?? 4138406020120,V.V).j.(VVVV pppNncnCN ????????? ???
)( l i g h tp h a s eBVV CNBN ???
Example:The balanced three-phase circuit (Y-Y),A phase load
is short -circuit (A).
anNn VV
?? ?0??
ANV
NnbnBN VVV
??? ??
NncnCN VVV ??? ??
bnV
?
cnV
?
anV
?
NnV
?
N
n
?
?
?a b
c
n
?anV
?
?
? ?
?
? ?
?
?
?A B
C
?
pZ
pZpZ N
bnV?
cnV?
anbn VV
?? ?? ???
abV
ancn VV ?? ?? caV??
BNV
?
CNV
?
a
b
c
1A
2A
3A
DP(3) If the switch is closed before t=0,(A1),(A2),
(A3) are 3.8 A,The switch is opened at t=0,find each
ampere-meters reading.
(A1)=2.2A;
(A2)=3.8A;
(A3)=2.2A.