Measurement of Power in a Three-phase AC Circuit Notes | EduRev

: Measurement of Power in a Three-phase AC Circuit Notes | EduRev

 Page 1


 
 
 
 
 
 
 
 
Module 
5 
 
Three-phase AC Circuits 
Version 2 EE IIT, Kharagpur 
Page 2


 
 
 
 
 
 
 
 
Module 
5 
 
Three-phase AC Circuits 
Version 2 EE IIT, Kharagpur 
 
 
 
 
 
 
 
 
Lesson 
20 
 
Measurement of Power 
in a Three-phase Circuit 
Version 2 EE IIT, Kharagpur 
Page 3


 
 
 
 
 
 
 
 
Module 
5 
 
Three-phase AC Circuits 
Version 2 EE IIT, Kharagpur 
 
 
 
 
 
 
 
 
Lesson 
20 
 
Measurement of Power 
in a Three-phase Circuit 
Version 2 EE IIT, Kharagpur 
In the previous lesson, the phase and line currents for balanced delta-connected load fed 
from a three-phase supply, along with the expression for total power, are presented. In 
this lesson, the measurement of total power in a three-phase circuit, both balanced and 
unbalanced, is discussed. The connection diagram for two-wattmeter method, along with 
the relevant phasor diagram for balanced load, is described.  
Keywords: power measurement, two-wattmeter method, balanced and unbalanced loads, 
star- and delta-connections.    
After going through this lesson, the students will be able to answer the following 
questions: 
1. How to connect the two-wattmeter to measure the total power in a three-phase circuit 
– both balanced and unbalanced? 
2. Also how to find the power factor for the case of the above balanced load, from the 
reading of the two-wattmeter, for the two types of connections – star and delta? 
Two-wattmeter Method of Power Measurement in a Three-
phase Circuit 
 
•
•
•
•
•
R
c
L
c
R
a
L
a
L
b
R
b
I
RN
1
I
YN
1
I
BN
1
W
2
Y 
B 
R 
W
1
V
RY
•
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 20.1 Connection diagram for two-wattmeter method of power measurement  
 in a three-phase balanced system with  star-connected load   
 
The connection diagram for the measurement of power in a three-phase circuit using 
two wattmeters, is given in Fig. 20.1. This is irrespective of the circuit connection – star 
or delta. The circuit may be taken as unbalanced one, balanced type being only a special 
case. Please note the connection of the two wattmeters. The current coils of the 
wattmeters, 1 & 2, are in series with the two phases, R & B , with the pressure or voltage 
Version 2 EE IIT, Kharagpur 
Page 4


 
 
 
 
 
 
 
 
Module 
5 
 
Three-phase AC Circuits 
Version 2 EE IIT, Kharagpur 
 
 
 
 
 
 
 
 
Lesson 
20 
 
Measurement of Power 
in a Three-phase Circuit 
Version 2 EE IIT, Kharagpur 
In the previous lesson, the phase and line currents for balanced delta-connected load fed 
from a three-phase supply, along with the expression for total power, are presented. In 
this lesson, the measurement of total power in a three-phase circuit, both balanced and 
unbalanced, is discussed. The connection diagram for two-wattmeter method, along with 
the relevant phasor diagram for balanced load, is described.  
Keywords: power measurement, two-wattmeter method, balanced and unbalanced loads, 
star- and delta-connections.    
After going through this lesson, the students will be able to answer the following 
questions: 
1. How to connect the two-wattmeter to measure the total power in a three-phase circuit 
– both balanced and unbalanced? 
2. Also how to find the power factor for the case of the above balanced load, from the 
reading of the two-wattmeter, for the two types of connections – star and delta? 
Two-wattmeter Method of Power Measurement in a Three-
phase Circuit 
 
•
•
•
•
•
R
c
L
c
R
a
L
a
L
b
R
b
I
RN
1
I
YN
1
I
BN
1
W
2
Y 
B 
R 
W
1
V
RY
•
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 20.1 Connection diagram for two-wattmeter method of power measurement  
 in a three-phase balanced system with  star-connected load   
 
The connection diagram for the measurement of power in a three-phase circuit using 
two wattmeters, is given in Fig. 20.1. This is irrespective of the circuit connection – star 
or delta. The circuit may be taken as unbalanced one, balanced type being only a special 
case. Please note the connection of the two wattmeters. The current coils of the 
wattmeters, 1 & 2, are in series with the two phases, R & B , with the pressure or voltage 
Version 2 EE IIT, Kharagpur 
coils being connected across Y R - and Y B - respectively. Y is the third phase, in 
which no current coil is connected.        
If star-connected circuit is taken as an example, the total instantaneous power 
consumed in the circuit is,  
 
N B N B N Y N Y N R N R
v i v i v i W
' ' ' ' ' '
· + · + · = 
Each of the terms in the above expression is the instantaneous power consumed for 
the phases. From the connection diagram, the current in, and the voltage across the 
respective (current, and pressure or voltage) coils in the wattmeter,  are  and 
. So, the instantaneous power measured by the wattmeter,  is,  
1
W
N R
i
'
N Y N R RY
v v v
' '
- =
1
W
()
N Y N R N R RY N R
v v i v i W
' ' ' '
- · = · =
1
    
Similarly, the instantaneous power measured by the wattmeter,  is,  
2
W
()
N Y N B N B BY N B
v v i v i W
' ' ' '
- · = · =
2
 
The sum of the two readings as given above is, 
() ()(
N B N R N Y N B N B N R N R N Y N B N B N Y N R N R
i i v v i v i v v i v v i W W
' ' ' ' ' ' ' ' ' ' ' ' '
+ ·) - · + · = - · + - · = +
2 1
  
Since,  or,  0 = + +
' ' ' N B N Y N R
i i i ( )
N B N R N Y
i i i
' ' '
+ = 
 
Substituting the above expression for in the earlier one,  
N Y
i
'
N Y N Y N B N B N R N R
v i v i v i W W
' ' ' ' ' '
· + · + · = +
2 1
  
If this expression is compared with the earlier expression for the total instantaneous 
power consumed in the circuit, they are found to be the same. So, it can be concluded that 
the sum of the two wattmeter readings is the total power consumed in the three-phase 
circuit, assumed here as a star-connected one. This may also be easily proved for delta-
connected circuit. As no other condition is imposed, the circuit can be taken as an 
unbalanced one, the balanced type being only a special case, as stated earlier. 
 
Version 2 EE IIT, Kharagpur 
Page 5


 
 
 
 
 
 
 
 
Module 
5 
 
Three-phase AC Circuits 
Version 2 EE IIT, Kharagpur 
 
 
 
 
 
 
 
 
Lesson 
20 
 
Measurement of Power 
in a Three-phase Circuit 
Version 2 EE IIT, Kharagpur 
In the previous lesson, the phase and line currents for balanced delta-connected load fed 
from a three-phase supply, along with the expression for total power, are presented. In 
this lesson, the measurement of total power in a three-phase circuit, both balanced and 
unbalanced, is discussed. The connection diagram for two-wattmeter method, along with 
the relevant phasor diagram for balanced load, is described.  
Keywords: power measurement, two-wattmeter method, balanced and unbalanced loads, 
star- and delta-connections.    
After going through this lesson, the students will be able to answer the following 
questions: 
1. How to connect the two-wattmeter to measure the total power in a three-phase circuit 
– both balanced and unbalanced? 
2. Also how to find the power factor for the case of the above balanced load, from the 
reading of the two-wattmeter, for the two types of connections – star and delta? 
Two-wattmeter Method of Power Measurement in a Three-
phase Circuit 
 
•
•
•
•
•
R
c
L
c
R
a
L
a
L
b
R
b
I
RN
1
I
YN
1
I
BN
1
W
2
Y 
B 
R 
W
1
V
RY
•
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 20.1 Connection diagram for two-wattmeter method of power measurement  
 in a three-phase balanced system with  star-connected load   
 
The connection diagram for the measurement of power in a three-phase circuit using 
two wattmeters, is given in Fig. 20.1. This is irrespective of the circuit connection – star 
or delta. The circuit may be taken as unbalanced one, balanced type being only a special 
case. Please note the connection of the two wattmeters. The current coils of the 
wattmeters, 1 & 2, are in series with the two phases, R & B , with the pressure or voltage 
Version 2 EE IIT, Kharagpur 
coils being connected across Y R - and Y B - respectively. Y is the third phase, in 
which no current coil is connected.        
If star-connected circuit is taken as an example, the total instantaneous power 
consumed in the circuit is,  
 
N B N B N Y N Y N R N R
v i v i v i W
' ' ' ' ' '
· + · + · = 
Each of the terms in the above expression is the instantaneous power consumed for 
the phases. From the connection diagram, the current in, and the voltage across the 
respective (current, and pressure or voltage) coils in the wattmeter,  are  and 
. So, the instantaneous power measured by the wattmeter,  is,  
1
W
N R
i
'
N Y N R RY
v v v
' '
- =
1
W
()
N Y N R N R RY N R
v v i v i W
' ' ' '
- · = · =
1
    
Similarly, the instantaneous power measured by the wattmeter,  is,  
2
W
()
N Y N B N B BY N B
v v i v i W
' ' ' '
- · = · =
2
 
The sum of the two readings as given above is, 
() ()(
N B N R N Y N B N B N R N R N Y N B N B N Y N R N R
i i v v i v i v v i v v i W W
' ' ' ' ' ' ' ' ' ' ' ' '
+ ·) - · + · = - · + - · = +
2 1
  
Since,  or,  0 = + +
' ' ' N B N Y N R
i i i ( )
N B N R N Y
i i i
' ' '
+ = 
 
Substituting the above expression for in the earlier one,  
N Y
i
'
N Y N Y N B N B N R N R
v i v i v i W W
' ' ' ' ' '
· + · + · = +
2 1
  
If this expression is compared with the earlier expression for the total instantaneous 
power consumed in the circuit, they are found to be the same. So, it can be concluded that 
the sum of the two wattmeter readings is the total power consumed in the three-phase 
circuit, assumed here as a star-connected one. This may also be easily proved for delta-
connected circuit. As no other condition is imposed, the circuit can be taken as an 
unbalanced one, the balanced type being only a special case, as stated earlier. 
 
Version 2 EE IIT, Kharagpur 
Phasor diagram for a three-phase balanced star-connected 
circuit 
 
V
BY
 
 
V
RY
V
YN
I
YN
V
BN
V
RN
I
RN
F 
F 
F 
F-30° 
I
BN 
30°+ F
30°
30°
Fig. 20.2 Phasor diagram for two-wattmeter method of power measurement  
 in a three-phase system with balanced star-connected load  
V
YB
30°
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The phasor diagram using the two-wattmeter method, for a three-phase balanced star-
connected circuit is shown in Fig. 20.2. Please refer to the phasor diagrams shown in the 
figures 18.4 &18.6b. As given in lesson No. 18, the phase currents lags the respective 
phase voltages by 
p
f f = , the angle of the load impedance per phase. The angle, f  is 
taken as positive for inductive load. Also the neutral point on the load ( ) is same as 
the neutral point on the source ( ), if it is assumed to be connected in star. The voltage 
at that point is zero (0). 
N '
N
The reading of the first wattmeter is,  
() ( ) () ? ? + ° · · · = + ° · · = · · = 30 cos 3 30 cos , cos
1 p p RN RY RN RY RN RY
I V I V I V I V W  
The reading of the second wattmeter is, 
() ( ) () ? ? - ° · · · = - ° · · = · · = 30 cos 3 30 cos , cos
2 p p BN BY BN BY BN BY
I V I V I V I V W  
The line voltage,  leads the respective phase voltage,  by , and the phase 
voltage,  leads the phase current,  by 
RY
V
RN
V ° 30
RN
V
RN
I f . So, the phase difference between & 
 is 
RY
V
RN
I ( ) f + ° 30 . Similarly, the phase difference between &  in the second case, 
can be found and also checked from the phasor diagram. 
BY
V
BN
I
Version 2 EE IIT, Kharagpur 
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