Courses

# 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
```
Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!