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ECE225 TWO-WATTMETER METHOD – UNBALANCED CASE SPRING 2008
−
a
b
c
A
B
C
b B
c C
a A
I
CA
−
bn
10
j
Ω
Ω
−j10Ω
∼
~
~
~
Z
~
∼
o V =127 0 V
o V =127 120 V
~
I cC
bB
I
~
aA
I
~
an
AB
I
~
BC
I
~
CA
n N ca
o V =127 −120 V
cn
∼
−
Z BC
−
Z AB
−
−
−
−
V
V
bc
ab
V
−
(a) Find
VAB ,
VBC , and
VCA geometrically.
VAB =
V
ab
V
BC
V
bc
V
CA
V
ca
(b) Find
Z
AB
Z
BC
, and
Z
CA
using the Y-to-∆ transformation:
ZAB =
Z
A
Z
B
Z
B
Z
C
Z
C
Z
A
Z
C
ZBC =
Z
A
Z
B
Z
B
Z
C
Z
C
Z
A
Z
A
Z
CA
Z
A
Z
B
Z
B
Z
C
Z
C
Z
A
Z
B
(c) Find P 3 φ.
P 3 φ =
(d) Find Q 3 φ.
Q 3 φ =
(e) Find
I
AB
I
BC
, and
I
CA
in polar and rectangular coordinates.
IAB =
V
AB
Z
AB
IBC =
V
BC
Z
BC
ICA =
V
CA
Z
CA
(f) Find
I
aA
I
bB
, and
I
cC
in rectangular and polar coordinates.
I
aA
I
AB
I
CA
I
bB
I
BC
I
AB
I
cC
I
CA
I
BC
(g) Find the value of a wattmeter W A
inserted between phases a and b (reference).
W
A
V
ab
I
aA
| cos(
6
V
ab
6
I
aA
(h) Find the value of a wattmeter WC inserted between phases c and b (reference).
W
C
V
cb
I
cC
| cos(^6
V
cb
I
cC
(i) Check that P 3 φ
= W
A
+ W
C
W
A
+ W
C
(j) Check whether Q 3 φ
3(W
C
− W
A
)? (Why or why not?)
3(WC − WA) =
(k) Finally, check that
S
3 φ
= P
3 φ
V
an
I
∗
aA
V
bn
I
∗
bB
V
cn
I
∗
cC
