Unbalanced Three-Phase Circuits

1. Lecture 5
Unbalanced Three-
Phase Systems
(Part II)

2. One method of solving an unbalanced three-wire star-connected
load by star-delta conversion is described in Lecture 4. But this
method is laborious and involved lengthy calculations. By using
Millman’s theorem, we can solve this type of problem in a much
easier way.
MILLMAN’S METHOD OF SOLVING UNBALANCED LOAD

3. Consider a number of impedances , , , …, which
terminate at common point N’ (see figure below). The other ends
of the impedances are connected to voltage sources numbered
as , , , … . Let N be any other point in the
network.
1
Z 2
Z 3
Z n
Z
1
V 2
V 3
V n
V
Vn
  Zn
Vi
Zi
Z3
Z2
Z1
V2 V3
V1
N
I1 I2 I3 I4 In
N’
MILLMAN’S METHOD OF SOLVING UNBALANCED LOAD

4. Application of Ohm’s law to the circuit leads to the following
equations:
1
'
1
1
!
Z
V
V
I
N
N
Z


2
'
2
2
Z
V
V
I
N
N
Z


3
'
3
3
Z
V
V
I
N
N
Z




n
N
N
n
Zn
Z
V
V
I
'


Vn
  Zn
Vi
Zi
Z3
Z2
Z1
V2 V3
V1
N
I1 I2 I3 I4 In
N’
MILLMAN’S METHOD OF SOLVING UNBALANCED LOAD

5. Next, by applying Kirchhoff’s current law to the currents at node N’,
we obtain the node equation
0
'
3
'
3
2
'
2
1
'
1









n
N
N
n
N
N
N
N
N
N
Z
V
V
Z
V
V
Z
V
V
Z
V
V

Solving this equation for VN’N, we get














 n
i i
n
i i
i
n
N
n
n
N
N
Z
Z
V
Z
Z
Z
Z
Z
Z
V
Z
V
Z
V
Z
V
V
1
1
3
2
1
3
3
2
2
1
1
'
1
1
1
1
1
1


MILLMAN’S METHOD OF SOLVING UNBALANCED LOAD

6. In terms of admittances, we can write













 n
i
i
n
i
i
i
n
n
n
N
N
Y
Y
V
Y
Y
Y
Y
Y
V
Y
V
Y
V
Y
V
V
1
1
3
2
1
3
3
2
2
1
1
'


where Yi =1/Zi. Thus, knowing the voltage the voltage drops V1, V2,
V3,…Vn and corresponding currents I1, I2, I3, …In, can be easily
determined.
Vn
  Yn
Vi
Yi
Y3
Y2
Y1
V2 V3
V1
N
I1 I2 I3 I4 In
N’

7. In words, Millman’s theorem states that:
If any number of admittances Y1, Y2, Y3, ...Yn meet at a
common point N’, and the voltages from another point N to
the free ends of these admittances are V1, V2, V3, ...Vn then
the voltage between points N’ and N is:













 n
i
i
n
i
i
i
n
n
n
N
N
Y
Y
V
Y
Y
Y
Y
Y
V
Y
V
Y
V
Y
V
V
1
1
3
2
1
3
3
2
2
1
1
'



8. The previous relationships enable us to formulate a method for the
analysis of unbalanced three-phase systems. The method consists
of three steps as follows:
( i ) Determine VN’N
( ii ) Calculate the currents IR, IY, IB, and IN.
( iii ) Find the phase and line voltages using Kirchhoff’s and
Ohm’s laws.

9. Application Of Millman’s Theorem To The Unbalanced 4-Wire
Y-Connected System
Consider an unbalanced wye (Y) load connected to a balanced
three-phase supply, as shown in the figure below. The system
contains conducting wires each of impedance ZL connecting the
source to the load, and a neutral wire of impedance connecting N
and N’. We wish to determine the phase voltages and their
corresponding phase currents.
ZL
ZL
ZL
ZL
VRN
VYN
VBN
ZR
ZB
ZY
N N’
IR
IN
IY
IB

10. To better see how we can apply Millman's Theorem to the solution of
this circuit, let us reconstruct the circuit into a circuit of parallel
branches consisting of a voltage source and a series resistance
(impedance), similar to the configuration shown in the earlier figure.
The reconstructed circuit is shown below.
ZL
N
VRN VBN
VYN
N’
IR IY IB IN
ZR ZY ZB
ZL ZL ZL
VN’N

11. Using the node N as the datum, we express the currents IR,
IY and IB in terms of phase voltages VRN, VYN, VBN and
node voltage VN’N
R
L
N
N
RN
R
Z
Z
V
V
I



'
Y
L
N
N
YN
Y
Z
Z
V
V
I



'
B
L
N
N
BN
B
Z
Z
V
V
I



'
L
N
N
L
N
N
N
Z
V
Z
V
I
'
'
0




Hence, we obtain the node equation
0
'
'
'
'











B
L
N
N
BN
Y
L
N
N
YN
R
L
N
N
RN
L
N
N
Z
Z
V
V
Z
Z
V
V
Z
Z
V
V
Z
V
Solving this equation for VN’N, we have
L
B
L
Y
L
R
L
B
L
BN
Y
L
YN
R
L
RN
N
N
Z
Z
Z
Z
Z
Z
Z
Z
Z
V
Z
Z
V
Z
Z
V
V
1
1
1
1
'













12. In terms of admittances, we have
L
B
Y
R
B
BN
Y
YN
R
RN
N
N
Y
Y
Y
Y
Y
V
Y
V
Y
V
V





 '
'
'
'
'
'
'
where
R
L
R
Z
Z
Y


1
'
Y
L
Y
Z
Z
Y


1
'
B
L
B
Z
Z
Y


1
'
Y
L
L
Z
Z
Y


1

13. An unbalanced four-wire, star-connected load has a balanced
voltage of 400 V, the loads are
Worked Example
 

 8
4 j
ZR
 

 4
3 j
ZY  

 20
15 j
ZB
Calculate the (i) line currents, and (ii) current in the neutral wire.
The conducting wires connecting the source to the load, each has an
impedance ZL = (0.09 + j0.16) Ω