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## Three-Phase System

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### ... * 1.8 Power in a Three Phase System * Power Calculation The three phase power is equal the sum of ... Q = 3 Vphase Iphase sin Apparent power, ... – PowerPoint PPT presentation

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Title: Three-Phase System

1
Chapter 1. Three-Phase System
2
• 1.1 Review of Single-Phase System
• The Sinusoidal voltage
• v1(t) Vm sin wt

i
v1
v2
AC generator
3
• 1.1 Review of Single-Phase System
• The Sinusoidal voltage
• v(t) Vm sin wt
• where
• Vm the amplitude of the sinusoid
• w the angular frequency in radian/s
• t time

4
The angular frequency in radians per second
5
• A more general expression for the sinusoid (as
shown in the figure)
• v2(t) Vm sin (wt q)
• where q is the phase

6
A sinusoid can be expressed in either sine or
cosine form. When comparing two sinusoids, it is
expedient to express both as either sine or
cosine with positive amplitudes. We can
transform a sinusoid from sine to cosine form or
vice versa using this relationship
sin (?t 180o) - sin ?t cos (?t 180o) -
cos ?t sin (?t 90o) cos ?t cos (?t 90o)
sin ?t
7
Sinusoids are easily expressed in terms of
phasors. A phasor is a complex number that
represents the amplitude and phase of a sinusoid.
v(t) Vm cos (?t ?)
Time domain
Phasor domain
Time domain Phasor domain

8
Time domain
v2(t) Vm sin (wt q)
v1(t) Vm sinwt
Phasor domain
or
or
9
1.1.1 Instantaneous and Average Power The
instantaneous power is the power at any instant
of time. p(t) v(t) i(t) Where v(t) Vm
cos (wt qv) i(t) Im cos (wt qi) Using
the trigonometric identity, gives
10
The average power is the average of the
instantaneous power over one period.
11
The effective value is the root mean square (rms)
of the periodic signal. The average power in
terms of the rms values is Where
12
1.1.2 Apparent Power, Reactive Power and Power
Factor The apparent power is the product of the
rms values of voltage and current. The
reactive power is a measure of the energy
exchange between the source and the load reactive
part.
13
The power factor is the cosine of the phase
difference between voltage and current. The
complex power
14
1.2 Three-Phase System In a three phase system
the source consists of three sinusoidal voltages.
For a balanced source, the three sources have
equal magnitudes and are phase displaced from one
another by 120 electrical degrees. A three-phase
system is superior economically and advantage,
and for an operating of view, to a single-phase
system. In a balanced three phase system the
power delivered to the load is constant at all
times, whereas in a single-phase system the power
pulsates with time.
15
1.3 Generation of Three-Phase
• Three separate windings or coils with terminals
R-R, Y-Y and B-B are physically placed 120o
apart around the stator.

16
V or v is generally represented a voltage, but to
differentiate the emf voltage of generator from
voltage drop in a circuit, it is convenient to
use e or E for induced (emf) voltage.
17
The instantaneous e.m.f. generated in phase R, Y
and B eR EmR sin wt eY EmY sin (wt
-120o) eB EmB sin (wt -240o) EmBsin (wt
120o)
17
18
Three-phase AC generator
IR
VR
ZR
ER
IN
EB
VB
VY
EY
ZB
ZY
IY
IB
19
Phase voltage
The instantaneous e.m.f. generated in phase R, Y
and B eR EmR sin ?t eY EmY sin (?t
-120o) eB EmB sin (?t -240o) EmBsin (?t
120o)
In phasor domain
120o
ER ERrms
0o
0o
EY EYrms
-120o
-120o
EB EBrms
120o
Magnitude of phase voltage
ERrms EYrms EBrms Ep
20
Three-phase AC generator
Line voltage
IR
VR
ERY
ZR
ER
IN
EB
VB
VY
EY
ZB
ZY
IY
IB
ERY ER - EY
21
Line voltage
ERY
-EY
120o
0o
-120o
ERY ER - EY
22
Three-phase AC generator
Line voltage
IR
VR
ZR
ER
IN
EB
VB
VY
EY
ZB
ZY
IY
IB
EYB
EYB EY - EB
23
Line voltage
120o
0o
-120o
-EB
EYB
EYB EY - EB
24
Three-phase AC generator
Line voltage
IR
VR
ZR
ER
EBR
IN
EB
VB
VY
EY
ZB
ZY
IY
IB
EBR EB - ER
25
Line voltage
EBR
120o
0o
-120o
-ER
For star connected supply, EL v3 Ep
EBR EB - ER
25
26
Phase voltages
It can be seen that the phase voltage ER is
reference.
ER Ep
0o
EY Ep
-120o
120o
EB Ep
120o
0o
Line voltages
-120o
27
Or we can take the line voltage ERY as reference.
27
28
Three-phase AC generator
Delta connected Three-Phase supply
IR
ERY
VR
ZR
ER
EB
VB
VY
EY
ZB
ZY
IY
IB
ERY
ER
Ep
0o
29
Three-phase AC generator
Delta connected Three-Phase supply
IR
VR
ZR
ER
EB
VB
VY
EBR
EY
ZB
ZY
IY
IB
EYB
For delta connected supply, EL Ep
30
Connection in Three Phase System
4-wire system (neutral line with impedance)
3-wire system (no neutral line )
4-wire system (neutral line without impedance)
3-wire system (no neutral line ), delta connected
system b) 3-wire system
system
31
4-wire system (neutral line with impedance)
Three-phase AC generator
IR
VR
ZR
ER
ZN
IN
EB
VB
VY
VN
EY
ZB
ZY
IY
IB
Voltage drop across neutral impedance
VN INZN
32
4-wire system (neutral line with impedance)
Three-phase AC generator
Applying KCL at star point
IR
VR
ZR
ER
ZN
IN
EB
VB
VY
VN
EY
ZB
ZY
IY
IB
IR IY IB IN
33
4-wire system (neutral line with impedance)
Three-phase AC generator
Applying KVL on R-phase loop
IR
VR
ZR
ER
ZN
IN
EB
VB
VY
VN
EY
ZB
ZY
IY
IB
34
4-wire system (neutral line with impedance)
Three-phase AC generator
Applying KVL on R-phase loop
IR
VR
ZR
ER
ZN
IN
VN
ER VR VN 0
ER IRZR VN 0
Thus
ER VN
IR
ZR
35
4-wire system (neutral line with impedance)
Three-phase AC generator
Applying KVL on Y-phase loop
IR
VR
ZR
ER
ZN
IN
EB
VB
VY
VN
EY
ZB
ZY
IY
IB
36
4-wire system (neutral line with impedance)
Three-phase AC generator
Applying KVL on Y-phase loop
EY VY VN 0
Thus
EY VN
IY
EY IYZY VN 0
ZY
ZN
IN
VY
VN
EY
ZY
IY
37
4-wire system (neutral line with impedance)
Three-phase AC generator
Applying KVL on B-phase loop
EB VB VN 0
Thus
EB VN
IB
EB IBZB VN 0
ZB
ZN
IN
EB
VB
VN
ZB
IB
38
4-wire system (neutral line with impedance)
Substitute Eq. 1.2, Eq.1.3, Eq. 1.4 and Eq. 1.5
into Eq. 1.1
IR IY IB IN
EB VN
EY VN
ER VN
VN

ZB
ZY
ZR
ZN
39
4-wire system (neutral line with impedance)
VN
40
4-wire system (neutral line with impedance)
VN is the voltage drop across neutral line
impedance or the potential different between load
star point and supply star point of three-phase
system.
VN
We have to determine the value of VN in order to
find the values of currents and voltages of star
41
Example
IR
EL 415 volt
VR
ZR 5 ?
ER
ZN 10 ?
IN
EB
VB
ZY 2 ?
VN
EY
ZB 10 ?
IY
IB
Find the line currents IR ,IY and IB. Also find
the neutral current IN.
42
3-wire system (no neutral line )
Three-phase AC generator
IR
VR
ZR
ER
ZN
IN
EB
VB
VY
VN
EY
ZB
ZY
IY
IB
43
3-wire system (no neutral line )
Three-phase AC generator
IR
VR
ZR
ER
VN
EB
VB
VY
EY
ZB
ZY
IY
IB
No neutral line open circuit , ZN 8
44
3-wire system (no neutral line )

VN

1

ZN
8

ZN
8
45
3-wire system (no neutral line )

VN

46
Example
IR
EL 415 volt
VR
ZR 5 ?
ER
EB
VB
ZY 2 ?
VN
EY
ZB 10 ?
IY
IB
Find the line currents IR ,IY and IB . Also find
the voltages VR, VY and VB.
47
3-wire system (no neutral line ),delta connected
Three-phase AC generator
IR
VR
ZR
ER
EB
VB
VY
EY
ZB
ZY
IY
IB
48
3-wire system (no neutral line ),delta connected
Three-phase AC generator
IR
Ir
ER
VRY
VBR
ZRY
ZBR
EB
Ib
ZYB
EY
Iy
IY
VYB
IB
49
3-wire system (no neutral line ),delta connected
Three-phase AC generator
IR
Ir
ERY
VRY
ER
VRY
VBR
ZRY
ZBR
EBR
VBR
EB
EY
Ib
ZYB
Iy
IY
VYB
IB
EYB
VYB
50
3-wire system (no neutral line ),delta connected
Phase currents
30o
Ir

-90o
Iy

150o
Ib

51
3-wire system (no neutral line ),delta connected
Three-phase AC generator
IR
Ir
Line currents
ERY
VRY
ER
VRY
IR
Ir
Ib
-
VBR
ZRY
ZBR
EBR
30o
150o
EL
VBR
-
EB

EY
Ib
ZBR
ZYB
Iy
IY
IY
Iy
Ir
-
VYB
EL
EYB
VYB
-90o
30o
IB
-

ZRY
52
3-wire system (no neutral line ),delta connected
Three-phase AC generator
IR
Ir
Line currents
ERY
VRY
ER
VRY
IB
Ib
Iy
-
VBR
ZRY
ZBR
EBR
150o
-90o
EL
VBR
-
EB

EY
Ib
ZYB
ZYB
Iy
IY
VYB
EYB
VYB
IB
53
Star to delta conversion
ZR
ZBR
ZRY
ZY
ZB
ZYB
54
Example
Use star-delta conversion.
IR
EL 415 volt
VR
ZR 5 ?
ER
EB
VB
ZY 2 ?
VN
EY
ZB 10 ?
IY
IB
Find the line currents IR ,IY and IB .
55
4-wire system (neutral line without impedance)
Three-phase AC generator
IR
VR
ZR
ER
ZN
0 ?
IN

EB
VB
VY
VN
EY
ZB
ZY
IR
IB
VN INZN IN(0) 0 volt
56
4-wire system (neutral line without impedance)
For 4-wire three-phase system, VN is equal to 0,
therefore Eq. 1.3, Eq. 1.4, and Eq. 1.5 become,
ER
ER VN
IR
ZR
EY
EY VN
IY
ZY
EB
EB VN
IB
ZB
57
Example
IR
EL 415 volt
VR
ZR 5 ?
ER
IN
EB
VB
ZY 2 ?
VN
EY
ZB 10 ?
IY
IB
Find the line currents IR ,IY and IB . Also find
the neutral current IN.
58
The instantaneous e.m.f. generated in phase R, Y
and B eR EmR sin wt eY EmY sin (wt
-120o) eB EmB sin (wt -240o) EmBsin (wt
120o)
59
1.4 Phase sequencesRYB and RBY
(a) RYB or positive sequence
sequence is produced when the rotor rotates in
the counterclockwise direction.
60
(b) RBY or negative sequence
sequence is produced when the rotor rotates in
the clockwise direction.
61
1.5 Connection in Three Phase System
1.5.1 Star Connection a) Three wire system
62
Star Connection b) Four wire system
63
64
1.5.2 Delta Connection
65
66
1.6 Balanced Load Connection in 3-Phase System
67
Wye-Connected Balanced Loads b) Three wire
system
Example
IR
EL 415 volt
VR
ZR 20 ?
ER
EB
ZY 20 ?
VB
VN
EY
ZB 20 ?
IY
IB
Find the line currents IR ,IY and IB . Also find
the voltages VR, VY and VB.
68
Wye-Connected Balanced Loads b) Three wire
system
VN 0 volt
VR ER
VY EY
VB EB
69
1.6.1 Wye-Connected Balanced Loadsa) Four wire
system
Example
IR
EL 415 volt
VR
ZR 20 ?
ER
IN
EB
ZY 20 ?
VB
VN
EY
ZB 20 ?
IY
IB
Find the line currents IR ,IY and IB . Also find
the neutral current IN.
70
1.6.1 Wye-Connected Balanced Loadsa) Four wire
system
For balanced load system, IN 0 and Z1 Z2
Z3
71
Wye-Connected Balanced Loads b) Three wire
system
72
Phase currents
Line currents
73
74
system
For unbalanced load system, IN ? 0 and Z1 ? Z2
? Z3
75
Phase currents
Line currents
76
1.8 Power in a Three Phase System
77
Power Calculation The three phase power is equal
the sum of the phase
powers P PR PY PB If the load is
balanced P 3 Pphase 3 Vphase
Iphase cos ?
78
1.8.1 Wye connection system I phase
I L and Real Power, P 3 Vphase Iphase
cos ? Reactive power,
Q 3 Vphase Iphase sin ? Apparent power,
S 3 Vphase Iphase
or S P jQ
79
1.8.2 Delta connection system
VLL Vphase P 3 Vphase
Iphase cos ?
80
1.9 Three phase power measurement
81
Power measurement In a four-wire system (3
phases and a neutral) the real power is
measured using three single-phase watt-meters.
82
Three Phase Circuit Four wire system, Each
phase measured separately
83
watt-meter connection
W
Current coil (low impedance)
voltage coil (high impedance)
83
84
a) Four wire system
Example
WR
IR
VR
?
ZR 5
30o
ER
EL 415 volt
IN
EB
VB
ZY 10
90o
EY
VN
?
WY
ZB 20
45o
?
IY
IB
WB
Find the three-phase total power, PT.
85
b) Three wire system
Example
WR
IR
VR
?
ZR 5
30o
ER
EL 415 volt
EB
ZY 10
90o
EY
VN
?
WY
ZB 20
45o
?
IY
IB
WB
Find the three-phase total power, PT.
86
b) Three wire system
Example
WR
IR
VR
?
ZR 5
30o
ER
EL 415 volt
EB
VB
ZY 10
90o
EY
VN
?
WY
ZB 20
45o
?
IY
IB
WB
Find the three-phase total power, PT.
87
Three Phase Circuit Three wire system,
• The three phase power is the sum of the two

88
Proving
The three phase power (3-wire system) is the sum
Instantaneous power pA vA iA pB vB iB pC
vC iC
pT pA pB pC vA iA vB iB vC iC
vA iA vB iB vC iC vA iA vB (-iA -iC)
vCiC
88
89
Proving
The three phase power (3-wire system) is the sum
Instantaneous power
pT vA iA vB (-iA iC) vCiC
(vA vB )iA (vC vB )iC
vAB iA vCBiC
pT pAB pCB
89
90
Power measurement In a four-wire system (3
phases and a neutral) the real power is
measured using three single-phase watt-meters.
In a three-wire system (three phases without
neutral) the power is measured using only two
single phase watt-meters. The watt-meters are
supplied by the line current and the line-to-line
voltage.
90