Title: ThreePhase System
1Chapter 1. ThreePhase System
2 1.1 Review of SinglePhase System
 The Sinusoidal voltage
 v1(t) Vm sin wt
i
v1
v2
Load
AC generator
3 1.1 Review of SinglePhase 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
4The 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
6A 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
7Sinusoids 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
8Time domain
v2(t) Vm sin (wt q)
v1(t) Vm sinwt
Phasor domain
or
or
91.1.1 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
10The average power is the average of the
instantaneous power over one period.
11The effective value is the root mean square (rms)
of the periodic signal. The average power in
terms of the rms values is Where
121.1.2 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.
13The power factor is the cosine of the phase
difference between voltage and current. The
complex power
141.2 ThreePhase 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 threephase
system is superior economically and advantage,
and for an operating of view, to a singlephase
system. In a balanced three phase system the
power delivered to the load is constant at all
times, whereas in a singlephase system the power
pulsates with time.
151.3 Generation of ThreePhase
 Three separate windings or coils with terminals
RR, YY and BB are physically placed 120o
apart around the stator.
16V 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.
17The 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
18Threephase AC generator
IR
VR
ZR
ER
IN
EB
VB
VY
EY
ZB
ZY
IY
IB
Threephase Load
19Phase 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
20Threephase AC generator
Line voltage
IR
VR
ERY
ZR
ER
IN
EB
VB
VY
EY
ZB
ZY
IY
IB
Threephase Load
ERY ER  EY
21Line voltage
ERY
EY
120o
0o
120o
ERY ER  EY
22Threephase AC generator
Line voltage
IR
VR
ZR
ER
IN
EB
VB
VY
EY
ZB
ZY
IY
IB
EYB
Threephase Load
EYB EY  EB
23Line voltage
120o
0o
120o
EB
EYB
EYB EY  EB
24Threephase AC generator
Line voltage
IR
VR
ZR
ER
EBR
IN
EB
VB
VY
EY
ZB
ZY
IY
IB
Threephase Load
EBR EB  ER
25Line voltage
EBR
120o
0o
120o
ER
For star connected supply, EL v3 Ep
EBR EB  ER
25
26Phase 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
27Or we can take the line voltage ERY as reference.
27
28Threephase AC generator
Delta connected ThreePhase supply
IR
ERY
VR
ZR
ER
EB
VB
VY
EY
ZB
ZY
IY
IB
Threephase Load
ERY
ER
Ep
0o
29Threephase AC generator
Delta connected ThreePhase supply
IR
VR
ZR
ER
EB
VB
VY
EBR
EY
ZB
ZY
IY
IB
EYB
Threephase Load
For delta connected supply, EL Ep
30Connection in Three Phase System
4wire system (neutral line with impedance)
3wire system (no neutral line )
4wire system (neutral line without impedance)
3wire system (no neutral line ), delta connected
load
StarConnected Balanced Loads a) 4wire
system b) 3wire system
DeltaConnected Balanced Loads a) 3wire
system
314wire system (neutral line with impedance)
Threephase AC generator
IR
VR
ZR
ER
ZN
IN
EB
VB
VY
VN
EY
ZB
ZY
IY
IB
Threephase Load
Voltage drop across neutral impedance
VN INZN
324wire system (neutral line with impedance)
Threephase AC generator
Applying KCL at star point
IR
VR
ZR
ER
ZN
IN
EB
VB
VY
VN
EY
ZB
ZY
IY
IB
Threephase Load
IR IY IB IN
334wire system (neutral line with impedance)
Threephase AC generator
Applying KVL on Rphase loop
IR
VR
ZR
ER
ZN
IN
EB
VB
VY
VN
EY
ZB
ZY
IY
IB
Threephase Load
344wire system (neutral line with impedance)
Threephase AC generator
Applying KVL on Rphase loop
IR
VR
ZR
ER
ZN
IN
VN
ER VR VN 0
ER IRZR VN 0
Thus
ER VN
IR
Threephase Load
ZR
354wire system (neutral line with impedance)
Threephase AC generator
Applying KVL on Yphase loop
IR
VR
ZR
ER
ZN
IN
EB
VB
VY
VN
EY
ZB
ZY
IY
IB
Threephase Load
364wire system (neutral line with impedance)
Threephase AC generator
Applying KVL on Yphase loop
EY VY VN 0
Thus
EY VN
IY
EY IYZY VN 0
ZY
ZN
IN
VY
VN
EY
ZY
IY
Threephase Load
374wire system (neutral line with impedance)
Threephase AC generator
Applying KVL on Bphase loop
EB VB VN 0
Thus
EB VN
IB
EB IBZB VN 0
ZB
ZN
IN
EB
VB
VN
ZB
IB
Threephase Load
384wire 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
394wire system (neutral line with impedance)
VN
404wire 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 threephase
system.
VN
We have to determine the value of VN in order to
find the values of currents and voltages of star
connected loads of threephase system.
41Example
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.
Threephase Load
423wire system (no neutral line )
Threephase AC generator
IR
VR
ZR
ER
ZN
IN
EB
VB
VY
VN
EY
ZB
ZY
IY
IB
Threephase Load
433wire system (no neutral line )
Threephase AC generator
IR
VR
ZR
ER
VN
EB
VB
VY
EY
ZB
ZY
IY
IB
Threephase Load
No neutral line open circuit , ZN 8
443wire system (no neutral line )
VN
1
ZN
8
ZN
8
453wire system (no neutral line )
VN
46Example
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.
Threephase Load
473wire system (no neutral line ),delta connected
load
Threephase AC generator
IR
VR
ZR
ER
EB
VB
VY
EY
ZB
ZY
IY
IB
Threephase Load
483wire system (no neutral line ),delta connected
load
Threephase AC generator
IR
Ir
ER
VRY
VBR
ZRY
ZBR
EB
Ib
ZYB
EY
Iy
IY
VYB
IB
Threephase Load
493wire system (no neutral line ),delta connected
load
Threephase AC generator
IR
Ir
ERY
VRY
ER
VRY
VBR
ZRY
ZBR
EBR
VBR
EB
EY
Ib
ZYB
Iy
IY
VYB
IB
EYB
VYB
Threephase Load
503wire system (no neutral line ),delta connected
load
Phase currents
30o
Ir
90o
Iy
150o
Ib
513wire system (no neutral line ),delta connected
load
Threephase 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
Threephase Load
523wire system (no neutral line ),delta connected
load
Threephase 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
Threephase Load
53Star to delta conversion
ZR
ZBR
ZRY
ZY
ZB
ZYB
54Example
Use stardelta conversion.
IR
EL 415 volt
VR
ZR 5 ?
ER
EB
VB
ZY 2 ?
VN
EY
ZB 10 ?
IY
IB
Threephase Load
Find the line currents IR ,IY and IB .
554wire system (neutral line without impedance)
Threephase AC generator
IR
VR
ZR
ER
ZN
0 ?
IN
EB
VB
VY
VN
EY
ZB
ZY
IR
IB
Threephase Load
VN INZN IN(0) 0 volt
564wire system (neutral line without impedance)
For 4wire threephase 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
57Example
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.
Threephase Load
58The 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)
591.4 Phase sequencesRYB and RBY
(a) RYB or positive sequence
VR leads VY, which in turn leads VB. This
sequence is produced when the rotor rotates in
the counterclockwise direction.
60 (b) RBY or negative sequence
VR leads VB, which in turn leads VY. This
sequence is produced when the rotor rotates in
the clockwise direction.
611.5 Connection in Three Phase System
1.5.1 Star Connection a) Three wire system
62Star Connection b) Four wire system
63Wye connection of Load
641.5.2 Delta Connection
65Delta connection of load
661.6 Balanced Load Connection in 3Phase System
67WyeConnected 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.
Threephase Load
68WyeConnected Balanced Loads b) Three wire
system
VN 0 volt
VR ER
VY EY
VB EB
691.6.1 WyeConnected 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.
Threephase Load
701.6.1 WyeConnected Balanced Loadsa) Four wire
system
For balanced load system, IN 0 and Z1 Z2
Z3
71WyeConnected Balanced Loads b) Three wire
system
721.6.2 DeltaConnected Balanced Loads
Phase currents
Line currents
731.7 Unbalanced Loads
741.7.1 WyeConnected Unbalanced LoadsFour wire
system
For unbalanced load system, IN ? 0 and Z1 ? Z2
? Z3
751.7.2 DeltaConnected Unbalanced Loads
Phase currents
Line currents
761.8 Power in a Three Phase System
77Power 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 ?
781.8.1 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
791.8.2 Delta connection system
VLL Vphase P 3 Vphase
Iphase cos ?
801.9 Three phase power measurement
81Power measurement In a fourwire system (3
phases and a neutral) the real power is
measured using three singlephase wattmeters.
82Three Phase Circuit Four wire system, Each
phase measured separately
83wattmeter connection
W
Current coil (low impedance)
voltage coil (high impedance)
83
84a) 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
Threephase Load
Find the threephase total power, PT.
85b) 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
Threephase Load
Find the threephase total power, PT.
86b) 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
Threephase Load
Find the threephase total power, PT.
87Three Phase Circuit Three wire system,
 The three phase power is the sum of the two
wattmeters reading
88Proving
The three phase power (3wire system) is the sum
of the two wattmeters reading
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
89Proving
The three phase power (3wire system) is the sum
of the two wattmeters reading
Instantaneous power
pT vA iA vB (iA iC) vCiC
(vA vB )iA (vC vB )iC
vAB iA vCBiC
pT pAB pCB
89
90Power measurement In a fourwire system (3
phases and a neutral) the real power is
measured using three singlephase wattmeters.
In a threewire system (three phases without
neutral) the power is measured using only two
single phase wattmeters. The wattmeters are
supplied by the line current and the linetoline
voltage.
90