Introduction to three phase power
•
•
Mains electrical power generating stations produce 3 emfs of equal
magnitude but separated in phase by 120°
The three generators may be in star or delta configuration:
a
a
~
V0120
V0
n
~
~
Vca
V0  120
Vab
~
b
c
c
Star point
~
~
Vbc
Positive (abc) sequence
For negative (or acb) sequence, Vbn  V0  120
Ref: Alexander and Sadiku, 5th Ed. Ch.12
1
b
3 phase line voltage
•
In the delta configuration the star point is not present and the voltages
are measured from one line to another. This line voltage is not the same
as the voltage measured from the star point in the star configuration.
V0
n
a
120°
V0
Line voltage
b
2
Vab =
3V0
Balanced Loads in 3 phase power transmission
Source
~
V0120
c
ia
a
z
a
V0
zL
n
n
~
ic
~
Load
V0  120
b
z
zL
c
zL
b
ib
z
Total current into star point of load =
ia  ib  ic 
V0  V0  120  V0120
z  zL
 1
3
1
3 
V0 1  
j 
j 
2 2
2 2 
 
0
z  zL
Hence (for balanced loads) there is zero current in the connection between the star
points of the generators and the loads, which is called the “neutral line”.
In practical 3 phase electrical power systems when the loads are not balanced a
(usually) small current flows in the neutral line.
3
3 phase power transmission
•
Distribution of 3 phase electrical power require only one conductor for
each phase, so the cables are in power transmission lines are in groups
of three.
4
Advantages of a Three-Phase Circuit
1. Most of the electric power is generated and distributed in
three-phase. When single phase or 2-phase inputs are
required, they are usually obtained from a 3-phase
system instead of generated independently.
2. The instantaneous power in a three-phase system can be
constant (time independent).
3. For the same amount of power, the three-phase system is
more economical than the single-phase. The amount of
wire required for a three-phase system is less than that
required for an equivalent single-phase system.
5
Balanced Three-Phase Voltages
Single phase electrical generator
•
A three-phase electrical generator consists of a rotating magnet
(rotor) surrounded by a stationary winding (stator).
A three-phase generator: terminals a and a’,
e.g., stands for ends of coils going into and 6
coming out of the page, respectively.
The generated voltages
Balanced Three-Phase Connection
•
Four possible connections
1. Y-Y connection (Y-connected source with
a Y-connected load)
2. Y-Δ connection (Y-connected source with
a Δ-connected load)
3. Δ-Δ connection
4. Δ-Y connection
7
Balanced Three-Phase Connection
• A balanced Y-Y system is a three-phase system with a
balanced y-connected source and a balanced y-connected load.
The 3 voltages have the same magnitude and frequency, and are out
of phase with each other by 120o.
See p.2
VL  3V p , where
V p  Van  Vbn  Vcn
VL  Vab  Vbc  Vca
8
Balanced Three-Phase Connection
Example
Calculate the line currents in the three-wire Y-Y system
shown below:
15  j 6  16.1621.8
Ans
I a  6.81  21.8 A
I b  6.81  141.8 A
I c  6.8198.2 A
9
Balanced Three-Phase Connection
• A balanced Y-Δ system is a three-phase system with a
balanced y-connected source and a balanced Δ-connected load.
IL: line current, Ip: phase current
KCL: I a  I AB  I CA  I AB  I AB   240
 I AB 1  0.5  j 0.886
 I AB 3  30
I L  3I p , where
I L  I a  Ib  Ic
I p  I AB  I BC  ICA
10
Balanced Three-Phase Connection
Example
A balanced abc-sequence Y-connected source with
( Van  10010 ) is connected to a Δ-connected load (8+j4) per
phase. Calculate the phase and line currents.
8.94426.57
Solution:
Using single-phase analysis,
Van
10010
Ia 

 33.54  16.57 A
Z / 3 2.98126.57
Other line currents are obtained using the abc phase sequence
Phase currents are obtained using result from p.10.
11
Balanced Three-Phase Connection
• A balanced Δ-Δ system is a three-phase system with a
balanced Δ -connected source and a balanced Δ -connected load.
12
Balanced Three-Phase Connection
25  36.87
Example
A balanced Δ-connected load having an impedance 20-j15  is
connected to a Δ-connected positive-sequence generator having
( Vab  3300 V). Calculate the phase currents of the load and
the line currents.
Ans:
The phase currents
I AB  13.236.87 A; I BC  13.2  81.13 A; I AB  13.2156.87 A
The line currents
I a  22.866.87 A; Ib  22.86  113.13 A; Ic  22.86126.87 A
I a  I AB 3  30 (see p.10)
13
Balanced Three-Phase Connection
• A balanced Δ-Y system is a three-phase system with a
balanced y-connected source and a balanced y-connected load.
14
Balanced Three-Phase Connection
47.1732
Example
A balanced Y-connected load with a phase impedance 40+j25  is
supplied by a balanced, positive-sequence Δ-connected source
with a line voltage of 210V. Calculate the phase currents. Use Vab
as reference.
Transform -connected source to a Y-connected source
V
Van  ab   30  121.2  30V
3
Answer
The phase currents
I AN  2.57  62 A;
I BN  2.57  178 A;
ICN  2.5758 A;
15