EE 221
CIRCUITS II
Chapter 12
Three-Phase Circuit
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THREE-PHASE CIRCUITS
CHAPTER 12
12.1
12.2
12.3
12.4
12.5
12.6
What is a Three-Phase Circuit?
Balanced Three-Phase Voltages
Balanced Three-Phase Connection
Power in a Balanced System
Unbalanced Three-Phase Systems
Application – Residential Wiring
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12.1 Advantages of Three-Phase Circuit?
1.
2.
3.
4.
Most of the electric power is generated and distributed
in three-phase.
The instantaneous power in a three-phase system can
be constant.
The amount of power generated by a three-phase
generator is more economical than that of a singlephase generator
The amount of wire required for a three-phase system
is less than that required for an equivalent singlephase system.
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Schematic Diagram of a Power System
US Electric Energy Generation Mix
Thermal Power Plants
Nuclear Power Plant
Natural Gas Power Plants
Hydro Power Plants
Renewable Power Plants
12.2 Balanced Three-Phase Voltages
|
A three-phase generator consists of a rotating
magnet (in the rotor) surrounded by stationary
windings (in the stator).
A three-phase generator
The generated voltages
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12.1 Definition of a Three-Phase Circuit
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It is a system with a generator consisting of three sources
having the same amplitude and frequency but out of phase
with each other by 120°.
Three sources
with 120° out
of phase
Four wired
system
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12.2 Balanced Three-Phase Voltages
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Two possible configurations:
Three-phase voltage sources: (a) Y-connected ; (b) Δ-connected
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12.2 Balance Three-Phase Voltages
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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.
y
y
|
a-b-c (or positive sequence)
a-c-b (or negative sequence)
A balanced load is one in which the phase
impedances are equal in magnitude and in phase
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12.2 Balance Three-Phase Voltages
Example 1
Determine the phase sequence of the set of
voltages.
van = 200 cos(ωt + 10°)
vbn = 200 cos(ωt − 230°)
vcn = 200 cos(ωt − 110°)
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12.2 Balanced Three-Phase Voltages
Solution:
The voltages can be expressed in phasor form as
Van = 200∠10° V
Vbn = 200∠ − 230° V
Vcn = 200∠ − 110° V
We notice that Van leads Vcn by 120° and Vcn in turn
leads Vbn by 120°.
Hence, we have an a-c-b sequence (or negative
sequence).
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12.3 Balanced Three-Phase Connection
|
Four possible connections
1.
2.
Y-Y connection (Y-connected source with
a Y-connected load)
Y-Δ connection (Y-connected source with
a Δ-connected load)
3.
Δ-Δ connection
4.
Δ-Y connection
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12.3 Balance Three-Phase Connection
• A balanced Y-Y system is a three-phase system with a
balanced Y-connected source and a balanced Yconnected load.
VL = 3V p , where
V p = Van = Vbn = Vcn
VL = Vab = Vbc = Vca
I n = −( I a + I b + I c ) = 0
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12.3 Balanced Three-Phase Connection
Example 2: Obtain the line currents
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12.3 Balanced Three-Phase Connection
• A balanced Y-Δ system is a three-phase system with a
balanced Y-connected source and a balanced Δconnected load.
I L = 3I p , where
I L = I a = Ib = Ic
I p = I AB = I BC = I CA
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12.3 Balanced Three-Phase Connection
Example 3:
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12.3 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.
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12.3 Balance Three-Phase Connection
Example 4:
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12.3 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.
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12.3 Balanced Three-Phase Connection
Example 5: let
Find the line currents
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12.4 Power in a Balanced System
P = 3V p I cos(θ ) = 3VL I cos(θ )
Q = 3V p I sin(θ ) = 3VL I sin(θ )
S = 3V p I = 3VL I
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12.3 Balanced Three-Phase Systems
Example 6
Determine the total complex power at the source and at the load
Ans
At the source:
Ss = (2087 + j834.6) VA
At the load:
SL = (1392 + j1113) VA
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12.3 BALANCED LOAD: EXAMPLE 7
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12.5 Unbalanced Three-Phase Systems
• An unbalanced system is due to unbalanced voltage
sources or an unbalanced load.
Ia =
VAN
V
V
, I b = BN , I c = CN ,
ZA
ZB
ZC
I n = −( I a + I b + I c )
•
The total power is not simply three times the power in one
phase but the sum of the powers in the three phases.
•
To calculate power in an unbalanced three-phase system
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requires that we find the power in each phase.
12.5 UNBALANCED THE-WIRE Y-CONNECTED
CIRCUIT
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Solve the problem again using nodal analysis.
12.5 UNBALANCED THE WIRE Δ CONNECTED
CIRCUIT
Calculate the line currents and total real and reactive power supplied by the source:
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Rework the problem using a simpler method
12.4 Power in a unbalanced System
Three Watt-Meter Method: PT = P1+P2+P3
Two Watt-Meter Method: PT = P1+P2
Special case: for a balanced load, the total reactive power may be computed
From the readings of the two wattmeters:
QT = 3 ( P2 − P1 )
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2-WATTMETER METHOD
The unbalanced load is supplied by a balanced source such that Vab = 208 0°
V with positive phase sequence. Calculate the reading of each wattmeter.
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Verify the answer by computing the power consumed by each resistor
12.6 Application – Residential Wiring
A 120/240 household power system
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12.6 Application – Residential Wiring
A typical wiring diagram of a room
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12.6 Application – Residential Wiring
+
7,600 V
-
Single-phase three-wire residential wiring
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12.6 Application – Residential Wiring
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UTILITY APPLICATION OF CAPACITORS
FOR VOLTAGE REGULATION AND
POWER TRANSFER
Capacitors are placed in series with high
Voltage transmission lines to improve voltage
Regulation and power transfer capability
Capacitor are also used in medium voltage power
Distribution lines to regulate voltage and reduce
power losses
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ENERGY METER (KILO-WATT-HOUR
METER)
Old electro-mechanical meter – records only
energy consumed over a time period using
mechanical dials.
New smart meter – records average power,
reactive power, power factor, power demand,
energy over a period of time, etc …
-Remote (wireless) reading
-Remote turn ON and OFF
-Communicates with the utility operator in
near-continuous basis
-Able to communicate with home appliances
such as HVAC thermostat….
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