Three phase circuits

10/01/2019 4.Three Phase Circuits - Google Docs
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INTRODUCTION

This chapter has been divided into following topics:
1) Basic concepts of Poly phase Circuits and its advantages
2) Generation of three phase supply
3) Star – Delta Connections (Balanced load)
4) Relationship between phase and line currents and voltages
5) Phasor diagrams
6) Measurement of power by two wattmeter method

POLYPHASE SYSTEMS

1) Poly phase alternators have two or more windings symmetrically spaced
around the armature. Such alternators produce as many independent
alternating voltages as the number of windings (Phase).
2) These voltages in the individual windings have the same magnitude and
frequency but they have definite phase difference. The amount of phase
difference depends upon the number of windings.
3) Thus, by using appropriate poly phase alternator, it is possible to generate
two phase and three phase ac and from these to obtain four, six, nine,
twelve phase AC depending upon the requirements. These systems are
collectively known as poly phase systems.

ADVANTAGES OF 3- Φ SYSTEM OVER 1- Φ SYSTEM

1) The output of a 3- φ machine is greater than that of 1- φ machine of the
same size.
2) The total power output of a 3- φ machine is not fluctuating as in the case
of a 1- φ machine. It has a higher efficiency.
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3) A 3- φ transmission needs less conducting material than a 1- φ
transmission line. Hence, transmission becomes economical.
4) 3- φ motors are self-starting, whereas 1- φ motors are not self-starting.
5) Power factor of a 1- φ motor is always lower than that of a 3- φ motor of
the same output and speed.
6) 1- φ supply can be obtained from 3- φ supply but 3- φ supply cannot be
obtained from 1- φ supply.

GENERATION OF THREE PHASE SUPPLY

Waveforms of 3- φ supply

Construction

1) The armature of the alternator consists of three single-turn rectangular
coils R1R2, Y1Y2 and B1B2 fixed to one another at angles of 120˚. The coils
are mounted on a common shaft and have same physical dimensions.
2) The ends of coil are connected to a pair of slip-rings carried on the shaft.
The coils are placed in the uniform magnetic field provided by the North
and South poles of the magnet.
3) The carbon brushes are pressed against the slip-rings to collect the
induced currents in the coils.

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Operation

1) Suppose the three coils are rotating in an anti-clockwise direction at
uniform speed. Because of this each coil will have its own generated e.m.f.
and current which will be alternating in nature.
2) As shown in the figure the plane of the coil R1R2 is perpendicular to the
magnetic field and hence no e.m.f. is generated in the coil. After 120˚ Y1Y2
will occupy this position and after 120˚ B1B2 will occupy this position.
3) This cycle continues and maximum value attained by every coil has 120˚
phase shift w.r.t e.m.f. in other coil.
4) If the instantaneous value of the e.m.f. generated in the coil R1R2 is
represented by,

then, the instantaneous values of the e.m.f.’s generated in the coil Y1Y2 and
B1B2 are,

5) Thus, we got three independent alternating voltages from 3- φ alternator
which have phase shift of 120˚. The phasor diagram for 3- φ system is as
show:

33

6) If we perform vector addition of these three voltages, it can be observed
that, the sum of three voltages at any instant is zero. Mathematically,

Important Definitions related to 3-φ System

1) Symmetrical System:-
A 3-φ system in which the three voltages are of same magnitude and frequency
and displaced by 120¢ª phase angle from each other is defined as
‘Symmetrical System’.
2) Phase Sequence:-
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The sequence in which the voltages in 3-φ reach their maximum positive
values is called ‘Phase Sequence’.
3) Balanced System:-
A 3-φ system is said to be balanced when,
a) All phase inpedances are identical.
b) Magnitude and phase angle of all phase impedances are identical.
c) All phases have same power factor.

Connections in Three Phase System

In 3- φ system we can have following connections:
a) Star or Wie (Y) connection:
b) Mesh or Delta (Δ) connection:
➢➢ Star Connection:-

1) In this connection, like terminals of the coil i.e. either starting terminals
R1, Y1, B1 or finishing terminals R2, Y2, B2 are connected together to form a
common point ‘N’ which is known as neutral point or star point.
2) This kind of connection is also known as ‘Four wire three phase system’.
This connection is shown in figure.
3) Assuming phase sequence R-Y-B all the line quantities and phase
quantities of star connection are shown in figure.
55

a) Line Voltage (VL) :
The potential difference between any two lines gives line to line voltage called
as line voltage.
For above fig. Line voltages are VL = VRY = VYB = VBR
b) Phase Voltage (Vph) :
The potential difference between any line and neutral point is called as phase
voltage.
For above fig. Phase voltages are Vph = VRN = VYN = VBN
c) Line Current (IL):
The current flowing through each line is called as line current.
For above fig. Line currents are IL = IR = IY = IB
d) Phase Current (Iph) :
The current flowing through each phase is called as phase current.
For above fig. phase currents are Iph = IRN = IYN = IBN
✓ Relation Between Line Quantities and phase Quantities for Star
Connected System
1) Consider a 3- φ star connected balanced system as shown in figure.
From figure,

But, and
Hence, …... (1)
2) To get the relation between line voltage and the phase voltage, plot the
vector diagram. Assuming phase sequence to be R-Y-B with lagging
power factor.
From fig. voltage across line R and line Y is
66

…... (2)
3) Vector Diagram for 3- φ star connected balanced system is shown in
below figure.
4) Consider
In draw
From vector diagram,

But ….... (3)
In

5) Substituting this in equation (3),
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Hence,
6) Thus, for a star connected connection we can conclude following points:
a) and
b) Line voltages are apart.
c) Line voltages are ahead of their respective phase voltages.
d) The angle between line currents and the corresponding line voltages
is with current lagging.
e) The angle between line currents and the corresponding line voltages
is with current leading.
✓ Expression for Power in 3- φ star connected system :
1) Active Power :
We know that for a 1-φ system active power is given by,
……(1)
Hence for a 3- φ system,
Total active power = 3 × (Power/phase)

Bus for star connected ckt.
88


Where φ is the angle between phase voltage and phase current.
2) Reactive Power :
Total reactive power is given by,

3) Apparent Power :
Total apparent Power is given by,

99

Delta Connection

1) In this connection finishing terminal of one coil is connected to starting
terminal of second coil and finishing of that to starting of next and so on
i.e. In Δ connection coils are connected to form a closed loop as shown in
fig. (1)
2) This kind of connection is also known as ‘three wire three phase system’.
This connection is shown in fig. (1)
3) Assuming phase sequence R-Y-B all the line quantities and phase
quantities of delta connection are shown in fig. (1)
a) Line Voltage (VL) :
The potential difference between any two lines gives line to line voltage called
as line voltage.
For above fig. Line voltages are VL = VRY = VYB = VBR
b) Phase Voltage (Vph) :
The potential difference between any line and neutral point is called as phase
voltage.
For above fig. Phase voltages are Vph = VR = VY = VB
c) Line Current (IL):
The current flowing through each line is called as line current.
For above fig. Line currents are IL = IR = IY = IB
1010

d) Phase Current (Iph) :
The current flowing through each phase is called as phase current.
For above fig. phase currents are Iph = IRY = IYB = IBR

✓ Relation Between Line Quantities and phase Quantities for Delta
Connected System:-
1. Consider a 3- φ delta connected balanced system as shown in figure.
From figure,

But,
2. To get the relation between line voltage and the phase voltage, plot the
vector diagram. Assuming phase sequence to be R-Y-B with lagging
power factor.
From fig.
…… (2)
3. Vector Diagram for 3- φ delta connected balanced system is shown
below figure.
1111


4. Consider
In draw
From vector diagram,

But …… (3)
In

5. Substituting this in equation (3),

Hence,
6. Thus, for a delta connected connection we can conclude following
points :
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a)
b) Line currents are 120 ˚ apart.
c) Line currents are 30˚ ahead of their respective phase currents.
d) The angle between line currents and the corresponding line voltages
is (30+ ) with current lagging.
e) The angle between line currents and the corresponding line voltages
is (30-φ) with current leading.
✓ Expression for Power in 3- φ delta connected system :
1) Active Power :

But for delta connected ckt.

Where φ is the angle between phase voltage and phase current.
2) Reactive Power :
Total reactive power is given by,

3) Apparent Power :
Total apparent Power is given by,

1313


Relationship between Power drawn by Star and Delta Connected Load

1) Let

2) For star connected system,
we have,

But

3) Now,

…..[From (1)]
….(2)
4) For delta connected system,
we have,
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But

5) Now,

….[From (3)]
….(4)
6) Thus, from . (2) and (4), we have

From this . we can conclude that for the same load, power drawn in delta
ckt. is three times more than the power drawn in star connected ckt.

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MEASUREMENT OF POWER IN THREE PHASE SYSTEM

Wattmeter

1. It is a device which is used to measure power drawn by single phase
circuit.. It consists of two coils (1) Current Coil (CC) and (2) Potential Coil
(PC)
2. Current coil senses the current and it is always connected in series with
the load. The resistance of the coil is very small and hence its cross
sectional area is large and it has less number of turns.
3. Potential coil is also known as ‘Pressure Coil’. This senses the voltage and
it is always connected across the supply terminals. The resistance of this
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coil is very high and hence its cross sectional area is small and it has more
number of turns
4. The symbol for wattmeter is as shown in fig. (1)

M Mains, C Common, L Line, V Voltage

5. The circuit connection for measurement of power of a single phase circuit
is shown in fig.(2)

6. Wattmeter reading is directly proportional to
a) Current (I) through the coil.
b) Voltage (V) across the coil.
c) Cosine of the angle between this voltage and current.
Thus, wattmeter reading is

Hence, wattmeter gives the direct reading of the power absorbed by the single
phase circuit.
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7. It is also used for measurement of power in 3-φ system. There are three
methods available for measuring power in 3-φ system. They are as follows
a) Three wattmeter method
b) Two wattmeter method
c) One wattmeter method

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TWO WATTMETER METHOD

Star Connection

1. The circuit diagram of two wattmeter method for star connected 3-
system is as shown above.
2. Current coil of each wattmeter is connected in any two lines and the
potential coils of both the wattmeter are connected to third line.
3. In above fig. current coils of wattmeter are connected in line-R
and line-Y respectively and their potential coils are shorted to line B.
4. For current through current coils is and voltage across potential
coil is . ….(1)
5. Similarly, for current through current coil is and voltage across
potential coil is .
1919

6. Hence reading is, = ….(2)
7. Now consider the phasor diagram for star ckt. Assuming phase sequence
R-Y-B and lagging p.f. we have,

1) From circuit diagram.

and
2) From vector Diagram we have,
Angle between

….(3)
….(4)
3) But for star connected balanced system,

2020

Hence from . (3) and (4), we have
….(5)
….(6)

4) Adding .’s (5) and (6) we get,

….(7)
5) But for star connected system,
Thus,
2121

i.e. the total power absorbed is equal to algebraic sum of the two wattmeter
readings.

DELTA CONNECTION
2222


1. The circuit diagram of two wattmeter method for delta connected 3-
system is as shown above.
2. Current coil of each wattmeter is connected in any two lines and the
potential coils of both the wattmeter are connected to third line.
3. In above fig. current coils of wattmeter and are connected in line-R
and line-Y respectively and their potential coils are shorted to line B.
4. For current through current coil is and voltage across potential coil
is . Hence reading is, ….(1)
5. Similarly, for current through current coil is and voltage across
potential coil is .
Hence reading is, ….(2)

6. Now consider the phasor diagram for delta ckt. Assuming phase sequence
R-Y-B and lagging p.f. we have,
2323


7. From circuit diagram.

and
8. From vector Diagram we have,
Angle between and is and angle between and is (30 + φ)
Thus eqn. (1) and (2) becomes,
W1 = VRB IR cos (30 - φ) ….(3)
W2 = VYB IY cos (30 + φ) ….(4)
9. But for delta connected balanced system,
VL = VRB = VYB and IL = IR = IY
Hence from eqn. (3) and (4), we have

….(5)
….(6)
10. Adding eqn.’s (5) and (6) we get,

2424


….(7)
11. But for delta connected system,
Thus,
i.e. the total power absorbed is equal to algebraic sum of the two
wattmeter readings.
EXPRESSION FOR POWER FACTOR USING TWO WATTMETER METHOD
1. When two wattmeter are used for measurement of power in 3-φ system,
the wattmeter readings are given by,
2. Adding (1) and (2) we get,

….(3)
3. Subtracting (2) from (1) we get,

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….(4)
4. From eqn. (3) and (4),

….(5)

5. But
….(6)
6. From the above expressions we can conclude following points
1) If , then the readings of two wattmeter are equal.
2) If , then the readings of both the wattmeter are positive.
2626

3) If , then the reading of wattmeter is zero.
4) If , then the reading of wattmeter is negative.

ADVANTAGES OF TWO WATTMETER METHOD

1. Availability of neutral point in star connected system is not compulsory,
for power measurement using two wattmeter method.
2. This method can be used for balanced as well as unbalanced systems.
3. It is very easy to determine of a system using two wattmeter
method.
4. Total volt-amperes can be obtained using two wattmeter readings for
balanced load.
5. Circuit connections are very simple.

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Three phase circuits

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