AC Circuits Three-Phase Circuits8 Balance Three-Phase Voltages • Balanced phase voltages are equal...
Transcript of AC Circuits Three-Phase Circuits8 Balance Three-Phase Voltages • Balanced phase voltages are equal...
AC CircuitsThree-Phase Circuits
Contents
• What is a Three-Phase Circuit?
• Balance Three-Phase Voltages
• Balance Three-Phase Connection
• Power in a Balanced System
• Unbalanced Three-Phase Systems
• Application – Residential Wiring
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A simple AC generators
Sinusoidal voltage sources
When a conductor rotates in a constant magnetic field, a sinusoidal wave is generated.
Single-phase system
When the loop is moving perpendicular to the lines of flux, the maximum voltage is induced.
When the conductor is moving parallel with the lines of flux, no voltage is induced.
Single-phase two-wire
Normal household: Single-phase three-wire
neutral
More pole pieces? More coils?
Three-phase circuit: A system produced by a generator consisting of three sources having the same amplitude and frequency but out of phase with each other by 120°.
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Polyphase
Three sources with 120° out
of phase
Four wired system
Advantages of a three-phase circuit
• Most of the electric power is generated and distributed in three-phase:
– Less/more than 3-phase required…taken from 3-phase
• The instantaneous power in a three-phase system can be constant (not pulsating):
– uniform power transfer and less vibration of three-phase machine
• The amount of power, the three-phase system is more economical that the single-phase.
• In fact, the amount of wire required for a three-phase system is less than that required for an equivalent single-phase system.
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Balance Three-Phase Voltages
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• A three-phase generator consists of a rotating magnet (rotor) surrounded by a stationary winding (stator).
A three-phase generator The generated voltages
It can supply power to both single-phase and three-phase load
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Balance Three-Phase Voltages
• Two possible configurations:
Three-phase voltage sources: (a) Y-connected ; (b) Δ-connected
Four wires Three wires
Phase voltage
Balanced voltages:
cnbnan
cnbnan
VVV
VVV
0
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Balance Three-Phase Voltages• Balanced phase voltages are equal in magnitude and are out of phase
with each other by 120°.
• The phase sequence is the time order in which the voltages pass through their respective maximum values.
• A balanced load is one in which the phase impedances are equal in magnitude and in phase
3
2
3
4
3
2
j
p
j
p
j
p
p
eVeV
eV
V
bn
cn
an
V
V
V
pV cnbnan VVV3
2
3
4
3
2
j
p
j
p
j
p
p
eVeV
eV
V
cn
bn
an
V
V
V abc/positive sequence
acb/negative sequence
abc/positive sequence
acb/negative sequence
Y321 ZZZZ ZZZZ cba
YZZ 3
Transform:
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Example
Example 1
Determine the phase sequence of the set of voltages.
)110cos(200
)230cos(200
)10cos(200
tv
tv
tv
cn
bn
an
Solution:
The voltages can be expressed in phasor form as
We notice that Van leads Vcn by 120° and Vcn in turn leads Vbn by 120°.Hence, we have an acb sequence.
V 110200V
V 230200V
V 10200V
cn
bn
an
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Balance 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
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Balance Y-Y systemA balanced Y-Y system is a three-phase system with a balanced y-connected
source and a balanced y-connected load.
cabcabL
cnbnanp
pL
V
V
VV
VVV
VVV
where, 3
LlSY ZZZZ
6
3
2
32
1
2
33
2
3
2
11
j
j
pp
VpejVpjVp
eVV
bnannbanab VVVVVLine voltage:
3
4
3
23
2
,jj
j
eee
,
aca
Y
an
Y
bnb
Y
ana III
Z
V
Z
VI
Z
VI
Line current (KVL):
,0- ,0 cbancba II IIII I 0 nNV
Voltage across the neutral line is zero: the neutral line can be removed (earth acting as the neutral line).
The same as the phase
current in Y-Y
Phasor diagram?
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Balanced Y-Y connection
Example 2
Calculate the line currents in the three-wire Y-Y system shown below:
A 2.9881.6I
A 8.14181.6I
A 8.2181.6I
Ans
c
b
a
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Balance Y- Connection
• A balanced Y-Δ system is a three-phase system with a balanced y-connected source and a balanced Δ-connected load.
CABCABp
cbaL
pL
I
I
II
III
III
where, 3
Line current
Phase current
Line voltage = the voltage across the load
abAB VV
Phase current
Δ
ABAB
Z
VI
Line current
6
1
3
4
3jj
a ee
ABABABCAAB IIIIII
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Balance Y- Connection: Example
Example 3
A balanced abc-sequence Y-connected source with V is connected to a Δ-connected load (8+j4) per phase. Calculate the phase and line currents.
Solution
Using single-phase analysis,
Other line currents are obtained using the abc phase sequence
10100Van
A 57.1654.3357.26981.2
10100
3/Z
VI an
a
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Balance - Connection
• A balanced Δ-Δ system is a three-phase system with a balanced Δ -connected source and a balanced Δ -connected load.
Line voltage = the phase voltage
abAB VV
Phase currentΔΔ
ABAB
Z
V
Z
VI ab
Line current
6
1
3
4
3j
j
a
e
e
AB
ABABCAAB
I
IIIII
pL II 3
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Balance - Connection: Example
Example 4
A balanced Δ-connected load having an impedance 20-j15 is connected to a Δ-connected positive-sequence generator having ( ). Calculate the phase currents of the load and the line currents.
Ans:
The phase currents
The line currents
V 0330Vab
A 87.1562.13IA; 13.812.13IA; 87.362.13I ABBCAB
A 87.12686.22IA; 13.11386.22IA; 87.686.22I cba
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Balance -Y Connection
• A balanced Δ-Y system is a three-phase system with a balanced Y-connected source and a balanced Y-connected load.
Can you work out the relationship between line current and phase current?
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Balance -Y Connection: Example
Example 5
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 210 V. Calculate the phase currents. Use Vab as reference.
Answer
The phase currents
A; 5857.2I
A; 17857.2I
A; 6257.2I
CN
BN
AN
Balanced three-phase: Summary of phase and line V/I
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Power in a Balanced System• Instantaneous power (Y-load, ZY = Z)
• Comparing the power loss in (a) a single-phase system, and (b) a three-phase system
phase-single ,2'2
2
L
Lloss
V
PRP phase- three,''
2
2
L
Lloss
V
PRP
• If same power loss is tolerated in both system, three-phase system use only 75% of materials of a single-phase system
3
2cos2 ,
3
2cos2 ,cos2
tVvtVvtVv pCNpBNpAN
3
2cos2 ,
3
2cos2 ,cos2
tIitIitIi pcpbpa
cos3 ppcCNbBNaANcba IVivivivpppp θIV QIVPjQP pppp sin ,cos , :phase oneFor *
ppIVS
Time independent
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Unbalanced Three-Phase Systems
• An unbalanced system is due to unbalanced voltage sources or an unbalanced load.
cban
C
CNc
B
BNb
A
ANa
IIII
,Z
VI ,
Z
VI ,
Z
VI
• To calculate power in an unbalanced three-phase system requires that we find the power in each phase.
• The total power is not simply three times the power in one phase but the sum of the powers in the three phases.
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Unbalanced Three-Phase Systems: Example
Example 6
Determine the total average power, reactive power, and complex power
at the source and at the load
AnsAt the source:Ss = -(2087 + j834.6) VA
Pa = -2087W
Pr = -834.6VAR
At the load:SL = (1392 + j1113) VA
Pa = 1392W
Pr = 1113VAR
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Application – Residential Wiring
A 120/240 household power system
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Application – Residential Wiring (2)
Single-phase three-wire residential wiring
A typical wiring diagram of a room