Three-phase Circuits
Balanced 3-phase systems
Unbalanced 3-phase systems
SKEE 1043 Circuit Theory
Dr. Nik Rumzi Nik Idris
1
Balanced 3-phase systems
Single-phase two-wire system:
•
Single source connected to a
load using two-wire system
Single-phase three-wire system:
•
•
Two sources connected to two
loads using three-wire system
Sources have EQUAL
magnitude and are IN PHASE
2
Balanced 3-phase systems
Balanced Two-phase three-wire system:
•
•
Two sources connected to two
loads using three-wire system
Sources have EQUAL
frequency but DIFFFERENT
phases
Balanced Three-phase four-wire system:
•
•
Three sources connected to 3
loads using four-wire system
Sources have EQUAL
frequency but DIFFFERENT
phases
3
Balanced 3-phase systems
WHY THREE PHASE SYSTEM ?
•
ALL electric power system in the world used 3-phase system to
GENERATE, TRANSMIT and DISTRIBUTE
•
Instantaneous power is constant – thus smoother rotation of
electrical machines
•
More economical than single phase – less wire for the same power
transfer
•
To pass SKEE 1043 – to be able to graduate !
4
Balanced 3-phase systems
Generation of 3-phase voltage: Generator
SEE VIDEO
5
Balanced 3-phase systems
Generation, Transmission and Distribution
6
Balanced 3-phase systems
Generation, Transmission and Distribution
7
Balanced 3-phase systems
Y and  connections
Balanced 3-phase systems can be considered as 3 equal single phase
voltage sources connected either as Y or Delta () to 3 single three loads
connected as either Y or 
SOURCE CONNECTIONS
LOAD CONNECTIONS
Y connected source
Y connected load
 connected source
 connected load
Y-Y
Y-
-Y
-
8
Balanced 3-phase systems
SOURCE CONNECTIONS
Source : Y connection
a
Van
v an ( t ) 
+

n
Vcn
b
Vbn
o
 Van  Vp  0
2 Vp cos(  t )
v bn ( t ) 
2 Vp cos(  t  120 )
V bn  Vp   120
v cn ( t ) 
2 Vp cos(  t  120 ) 
Vcn  Vp 120
o
o
o
RMS phasors !
c
240o
120o
Van
o
Vbn
Vcn
9
Balanced 3-phase systems
Source : Y connection
Vcn  Vp 120
a
Van
SOURCE CONNECTIONS
+

o
120o
n
Vcn
Vbn
V an  Vp  0
120o
120o
b
c
V bn  Vp   120
o
Phase sequence : Van leads Vbn by 120o and Vbn leads Vcn by 120o
This is a known as abc sequence or positive sequence
10
o
Balanced 3-phase systems
SOURCE CONNECTIONS
Source : Y connection
a
Van
v an ( t ) 
+

v cn ( t ) 
o
 Van  Vp  0
2 Vp cos(  t )
2 Vp cos(  t  120 ) 
o
Vcn  Vp   120
o
n
Vcn
v bn ( t ) 
o
2 Vp cos(  t  120 ) V bn  Vp  120
b
Vbn
RMS phasors !
c
120o
Van
o
240o
Vcn
Vbn
11
Balanced 3-phase systems
Source : Y connection
V bn  Vp  120
a
Van
SOURCE CONNECTIONS
+

o
120o
n
Vcn
Vbn
V an  Vp  0
120o
b
c
120o
Vcn  Vp   120
o
Phase sequence : Van leads Vcn by 120o and Vcn leads Vbn by 120o
This is a known as acb sequence or negative sequence
12
o
Balanced 3-phase systems
SOURCE CONNECTIONS
Source :  connection
a
v bc ( t ) 
Vab
Vca
v ab ( t ) 
+

b
v ca ( t ) 
o
 Vab  Vp  0
2 Vp cos(  t )
o
2 Vp cos(  t  120 ) Vbc  Vp   120
2 Vp cos(  t  120 )
o
Vca  Vp 120
o
o
Vbc
RMS phasors !
c
120o
240o
Vab
Vbc
Vca
13
Balanced 3-phase systems
LOAD CONNECTIONS
 connection
Y connection
a
a
Z1
Zb
Zc
n
b
Z2
b
Z3
Za
c
c
Balanced load:
Z 1 = Z 2 = Z3 = ZY
Z a = Z b = Zc = Z
ZY 
Z
3
Unbalanced load: each phase load may not be the same.
14
Balanced 3-phase systems
Ia
A
a
Van
Vcn
V an  Vp  0
+

In
N
n
V cn  V p  120
c
Vbn
Ib 
Ic 
o
V bn  V p   120
ZY
Ib
Ia 
Balanced Y-Y Connection
Vp  0
b Ic
ZY
o
C
V ab  V an  V nb
ZY
 V p  0  V p  60
o
V p   120
o
ZY
V p  120
Phase
voltages
ZY
B
o
o
line
currents
o

3 V p  30
o
Vbc  Vbn  Vnc

3 Vp   90
ZY
 Ia  Ib  Ic  In  0
The wire connecting n and N can be removed !
V ca  V cn  Vna

o
3 Vp  150
o
o
line-line
voltages
OR
Line
voltages
15
Balanced 3-phase systems
Balanced Y-Y Connection
V ab  V an  V nb
 V p  0  V p  60
o

3 V p  30
o
o
16
Balanced 3-phase systems
Balanced Y-Y Connection
V ab  V an  V nb
 V p  0  V p  60
o

3 V p  30
o
V cn
V an
o
Vbn
17
Balanced 3-phase systems
Balanced Y-Y Connection
V ab  V an  V nb
 V p  0  V p  60
o

3 V p  30
o
o
V an
Vnb
18
Balanced 3-phase systems
Balanced Y-Y Connection
V ab  V an  V nb
 V p  0  V p  60
o

3 V p  30
o
V an
o
Vnb
Vbn
Vbc  Vbn  Vnc

3 Vp   90
o
Vnc
19
Balanced 3-phase systems
Balanced Y-Y Connection
Vna
V ab  V an  V nb
 V p  0  V p  60
o

3 V p  30
V cn
o
V an
o
Vnb
Vbn
Vbc  Vbn  Vnc

3 Vp   90
o
Vnc
V ca  V cn  Vna

3 Vp  150
o
20
Balanced 3-phase systems
V ca
V ab  V an  V nb
 V p  0  V p  60
o

3 V p  30
Balanced Y-Y Connection
o
V ab
30
o
V cn
o
30
V an
Vbn
Vbc  Vbn  Vnc

30
3 Vp   90
3 Vp  150
o
o
Vbc
V ca  V cn  Vna

o
o
VL 
where
VL  V ab  V bc  V ca
3 Vp
and V p  V an  Vbn  V cn
Line voltage LEADS phase voltage by 30o
21
Balanced 3-phase systems
Balanced Y-Y Connection
For a balanced Y-Y connection, analysis can be performed using an
equivalent per-phase circuit: e.g. for phase A:
Ia
A
a
Van
Vcn
+

ZY
In=0
N
n
Ib
c
Vbn
b Ic
ZY
B
ZY
C
22
Balanced 3-phase systems
Balanced Y-Y Connection
For a balanced Y-Y connection, analysis can be performed using an
equivalent per-phase circuit: e.g. for phase A:
Ia
A
a
Van
+

ZY
N
n
Ia 
V an
ZY
Based on the sequence, the other line currents can be
obtained from:
I b  I a   120
o
I c  I a  120
o
23
Balanced 3-phase systems
Balanced Y- Connection
Ia
a
Van
A
+

Z
n
Vcn
Ib
c
Vbn
V ab 
3 V p  30
o
B
b Ic
I AB 
3 Vp   90
3 Vp  150
 V CA
C
I CA
V cn  V p  120
V AB
Using KCL,
o
o
o
o
Ia  I AB  ICA
 I AB (1  1 120 )
o
Z
 I AB
I BC 
 VBC
V ca 
V bn  V p   120
Z
I AB Z

o
IBC
 V AB
Vbc 
V an  Vp  0
ICA 
V BC
Z
V CA
Z
Phase
currents
3   30
o
Ib  IBC  I AB
 IBC (1  1 120 )
o
 IBC
3   30
o
Ic  ICA
3   30
o
24
Balanced 3-phase systems
Balanced Y- Connection
Ic
ICA
30
o
I AB
30
Ib
Ia  I AB  ICA
o
30
o
 I AB (1  1 120 )
o
 I AB
IBC
Ib  IBC
IL 
IL  I a  I b  I c
o
 IBC (1  1 120 )
o
3 Ip
 IBC
where
3   30
Ia
 I AB
and
3   30
o
Ip  I AB  I BC  I CA o
Ic  ICA 3   30
Phase current LEADS line current by 30o
25
Balanced 3-phase systems
Balanced - Connection
Ia
a
V ab  V p  0
A
Vca
Vab
Ib
+

c
Vbc
V ab  V AB
Z
B
b Ic
I AB 
V bc  V p   120
Z
I AB Z

C
I CA
Vbc  VBC
V ca  V CA
ICA 
V cn  V p  120
o
o
IBC
V AB
Using KCL,
Ia  I AB  ICA
 I AB (1  1 120 )
o
Z
 I AB
I BC 
o
V BC
Z
V CA
Z
Phase
currents
3   30
o
line
currents
Ib  IBC  I AB
 IBC (1  1 120 )
o
 IBC
3   30
o
Ic  ICA
3   30
o
26
Balanced 3-phase systems
Balanced - Connection
Ia
a
V ab  V p  0
A
Vca
Z
Ib
+

c
Vab
Vbc
b Ic
B
V bc  V p   120
Z
I AB Z

o
C
I CA
V cn  V p  120
o
o
IBC
Alternatively, by transforming the  connections to the equivalent Y
connections per phase equivalent circuit analysis can be performed.
27
Balanced -Y Connection
Balanced 3-phase systems
Ia
A
a
V ab  V p  0
Vca
N
Ib
Vbc
V bc  V p   120
ZY
Loop1
+

c
Vab
ZY
V ca  V p  120
ZY
B
b Ic
o
o
o
C
How to find Ia ?
 Ia  Ib 
Loop1 - V ab  Z Y I a  Z Y I b  0
Since circuit is balanced, Ib = Ia-120o
V ab
ZY
 I a  I b  I a (1  1 (  120 ))
o
 Ia
Therefore
Ia 
Vp
3
  30
3  30
o
o
ZY
28
Balanced -Y Connection
Balanced 3-phase systems
Ia
A
a
V ab  V p  0
Vca
N
Ib
+

c
Vbc
V bc  V p   120
ZY
Vab
b Ic
ZY
B
o
V ca  V p  120
ZY
o
o
C
How to find Ia ? (Alternative)
Transform the delta source connection to an equivalent Y and then
perform the per phase circuit analysis
29