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EGR 272 – Circuit Theory II
Lecture #12
Read: Chapter 11 in Electric Circuits, 6th Edition by Nilsson
Generator and line impedances
Sometimes impedances in the generator and the lines are considered in 3-phase
circuits. This will result in power losses in each line and in reduced load voltages.
Example: The 4-wire Y-Y system below has a balanced generator with 720 V RMS
and an acb phase sequence. Determine IaA, Van, VAN, and the power loss in each line.
a
+
1
A
+
j2
+
j3
2
5
+
j10
n
N
-
-
+
+
j10
b
+
1
B
+
j2
1
Generator
j2
+
j3
2
c
+
j10
5
C
+
2
j3
Line
+
5
Load
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Lecture #12
EGR 272 – Circuit Theory II
- system
Recall for a  generator that VL = Vp. So analyzing the - system is similar to
analyzing the Y- system except that the line voltages are more easily found.
Example: Determine the three line currents for a - system that has a balanced
generator with 240 V RMS and a negative phase sequence. The loads are as follows:
ZAB = 3+j4, ZBC = 3-j4, and ZCA = 2+j2.
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Lecture #12
EGR 272 – Circuit Theory II
 Load with line impedances
Including line impedances with a  load makes the analysis more difficult. A good
way to approach the problem is to use a -Y conversion to change the  load into a
Y load.
Delta-to-Wye (-Y) and Wye-to-Delta (Y-) Transformations
In EGR 271 -Y and Y- transformations were used with resistive circuits. These
transformations can also be used with circuits consisting of AC impedances. Recall
that the transformation equations are derived based on a specific labeling of the
impedances, so the equations below are somewhat useless without the
corresponding figures.
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EGR 272 – Circuit Theory II
Lecture #12
Delta-to-Wye (-Y) and Wye-to-Delta (Y-) Transformations
a
b
a
Z1
b
Zc
Z2
Zb
Za
Z3
c
Wye Circuit
Y -  Conversion Equations :
c
Delta Circuit
 - Y Conversion Equations :
Za 
Z1  Z 2  Z 2  Z 3  Z3  Z1
Z1
Z1 
Zb  Z c
Za  Z b  Zc
Zb 
Z1  Z 2  Z 2  Z 3  Z3  Z1
Z2
Z2 
Zc  Z a
Za  Z b  Zc
Zc 
Z1  Z 2  Z 2  Z 3  Z3  Z1
Z3
Z3 
Za  Z b
Za  Z b  Zc
Special case: If the load is balanced, these equations reduce to:
Z  3  ZY (balanced load)
ZY 
Z
(balanced load)
3
4
Lecture #12
EGR 272 – Circuit Theory II
Example: Determine the three line currents in a 3 Y- circuit described as follows:
• The Y generator is balanced with an abc phase sequence and Van = 480 V
• Each of the 3 lines (between source and load) has an impedance of 2 + j4 ohms
• The  load is balanced where each of the three loads have an impedance of 60 +
j90 ohms
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Lecture #12
EGR 272 – Circuit Theory II
Power Calculations in 3 Circuits
Total power delivered = (power delivered to each phase)
Or
PTotal   P where P  V I cos( )
and V , I , and  are the voltage, current, and phase angle for each phase
and  is the phase angle of the impedance (or   V -  I )
Note: Power can be calculated, as it would be for any AC circuit. For example, total
power could be found by finding the power to the resistive portion to each load.
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Lecture #12
EGR 272 – Circuit Theory II
Example: Find the total power delivered to the Y-Y circuit analyzed last class
(4-wire Y-Y system has a balanced generator with Van = 480 V and a positive
phase sequence with ZAN = ZBN = 2 + j2 and ZCN = 2 - j2).
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Lecture #12
EGR 272 – Circuit Theory II
Example: Find the total power delivered to the Y-Y circuit analyzed last class
(balanced system with Van = 240 V, a negative phase sequence, and with
impedances as follows: ZAB = 6 + j8, ZBCN = 6 – j8, and ZCA = 6).
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Lecture #12
EGR 272 – Circuit Theory II
Measuring Power with Wattmeters:
A wattmeter is a piece of equipment that measures average power, P, in watts. A
wattmeter has connections for both current and voltage, as shown below on the
left (Electric Circuits, 9th Ed., by Nilsson). Note that the positive side of the
current coil and the positive side of the voltage coil are labeled + or +. The
wattmeter shown below on the right shows how a wattmeter might be connected
in a circuit.
a
IaA
+
+
W1
+
+
Van
Van
-
n
9
EGR 272 – Circuit Theory II
Lecture #12
Wattmeter reading:
A wattmete r reads real (or average) power, P, in watts (W).
I
+
+
W1
+
V
-
It reads the real part of the complex power, S  V  I
*
where V and I are the voltage and current at the wattmeter terminals .
so
 
P1  W1  Re V  I
*
or
P1  W1  V  I cos( ), where  is the phase difference between V and I
The 2-wattmeter method and the 3-wattmeter method:
Two common methods for measuring power are the 2-wattmeter method and the
3-wattmeter method. In the 3-wattmeter method, all negative voltage connection
on each of the wattmeters is common (typically on the neutral line). In the 2wattmeter method, the positive voltage terminal on two wattmeters is connected to
any two of the lines and both negative terminals are connected to the third line. It
can be proven that total power is the sum of the wattmeter readings in either
method.
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Both methods are illustrated on the following page.
EGR 272 – Circuit Theory II
Lecture #12
 
The 3-wattmeter method:
IaA
a
+
+
Van
-
A
W1+
+
Van
Vcn
-
Vbn
W2
+
+
+
b
ZCN
ZBN
Vbn
IbB
N
-
+
+
IcC
W3
+
c
Vcn
+
+


*


*
 
ZAN
-
-
n
W1  Re  V an  IaA

W2  Re V bn  I bB

B
C
 
*

W3  Re V cn  IcC 


PT  W1  W2  W3
The 2-wattmeter method:
IaA
a
+
+
Van
-
A
+
Vac
N
Vbn
-
+
+
IbB
b
ZCN
+
WB
+
Vbc
-
 
 I  
B
*
*
bc
bB
PT  WA  WB
ZBN
+

W  Re V
WA  Re V ac  I aA
ZAN
-
n
Vcn
c
WA+
B
C
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EGR 272 – Circuit Theory II
Lecture #12
Example: Determine the reading for each wattmeter below and the total power
absorbed by the load if the circuit has a balanced generator with Van = 480V, a
positive phase sequence, and impedances ZAN = 6+j8, ZBN = 8+j6, and ZCN = 5-j5.
IaA
a
+
A
+
WA
+
Vac
+
n
-
Vcn
N
Vbn
+
+
IbB
b
c
ZAN
Van
ZBN
ZCN
+
+
WB
+
Vbc
-
B
C
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