Download balanced three–phase circuits

JAMES W. NILSSON
&
SUSAN A. RIEDEL
ELECTRIC
CIRCUITS
EIGHTH EDITION
CHAPTER 11
BALANCED
THREE–PHASE
CIRCUITS
© 2008 Pearson Education
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CONTENTS
11.1 Balanced Three-Phase Voltages
11.2 Three-Phase Voltage Sources
11.3 Analysis of the Wye-Wye Circuit
11.4 Analysis of the Wye-Delta Circuit
11.5 Power Calculations in Balanced ThreePhase Circuits
11.6 Measuring Average Power in ThreePhase Circuits
© 2008 Pearson Education
Constant rms vtg. must be supplied
whether lightly loaded, as at 3:00 am, or
heavily loaded, as at midafternoon on a hot,
humid day.
One technique for maintaining vtg. levels
on a utility sys. is to place capacitors at
strategic locations in the distribution network.
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11.1 Balanced Three-Phase Voltages
A basic three-phase circuit
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3 vtgs.:
A-phase vtg.
B-phase vtg.
C-phase vtg.
11.1 Balanced Three-Phase Voltages
The abc (positive)
sequence
The abc (negative)
sequence
Phasor diagrams of a balanced set of three-phase voltages
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11.2 Three-Phase Voltage Sources
A three-phase voltage source is
a generator with three separate
windings distributed around the
periphery of the stator.
Rotor: electromagnet driven by
steam or gas turbine.
Stator is the stationary part of a
rotor system and is winded by 3
seperated windings.
a three-phase voltage source
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11.2 Three-Phase Voltage Sources
The two basic connections of an ideal
three-phase source.
A Y-connected source
A ∆-connected source
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11.2 Three-Phase Voltage Sources
A model of a three-phase source with
winding impedance:
(a) A Y-connected source
(b) A ∆-connected source
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Basic ckt configuration between S. and load
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11.3 Analysis of the Wye-Wye Circuit
: internal impedance of each phase
in generator
: line impedance
: load impedance
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See:
By the way,
=0
(balanced 3-phase vtg.)
Therefore,
Also,
Now turn to the effect that balanced conds. have 3 line ct.
11.3 Analysis of the Wye-Wye Circuit
A single-phase equivalent circuit
A single-phase equivalent circuit is used to
calculate the line current and the phase
voltage in one phase of the Y-Y structure. The
a-phase is normally chosen for this purpose.
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Caution: ct. in neutral conductor is IaA , which is not
the same as ct. in neutral conductor, that is,
(IaA,or IbB,or IcC),
We can find any vtg. easily.
11.3 Analysis of the Wye-Wye Circuit
Phasor diagrams showing the relationship
between line-to-line and line-to-neutral
voltages in a balanced system.
The abc sequence
The acb sequence
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Above expressions explain the following:
Terminology:
Line vtg. (VL) refers to vtg. across any pair of lines.
Phase vtg. (VΦ) refers to vtg. across a single phase.
Line ct. (IL) refers to ct. in a single line.
Phase ct.(IΦ) refers to ct. in a single phase.
See:
Line vtg. = phase vtg.
Line ct. = phase ct.
Balanced 3 phase Y-connected generator:
- impedance of gen.:(0.2 + j0.5)Ω
- internal vtg. :120 V/φ
Balanced 3 phase Y-connected impedance:
(39 + j28)Ω/φ
Line impedance: (0.8 + j1.5)Ω
a phase vtg. : reference phasor vtg.
After the Δ load has been replaced by its Y equivalent, the a-phase
can be modeled by the single phase equivalent circuit shown in Fig.
11.11.
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11.4 Analysis of the Wye-Delta Circuit
A circuit used to establish the relationship between
line currents and phase currents in a balanced ∆ load
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11.4 Analysis of the Wye-Delta Circuit
Phasor diagrams showing the relationship
between line currents and phase currents
in a ∆-connected load.
The positive sequence
The negative sequence
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Y-connected S.:
- impedance of gen.:(0.2 + j0.5)Ω
- internal vtg. :120 V/φ
Δ-connected load impedance:
(118.5 + j85.8)Ω/φ
cf: Y-connected impedance:
(39 + j28)Ω/φ
Line impedance: (0.3 + j0.9)Ω
a phase vtg. : reference phasor vtg.
Y-to-Y ckt
See:
Line vtg. = phase vtg.
Line ct. = phase ct.
11.5 Power Calculations in Balanced
Three-Phase Circuits
Average Power in a Balanced Wye Load
A balanced Y load used to introduce average
power calculations in three-phase circuits
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See: - all phasor ct. & vtg. are rms value .
- in this sys.,
11.5 Power Calculations in Balanced
Three-Phase Circuits
VL
PT  3
I L cos 
3
 3VL I L cos 
Total real power in a balanced three-phase load
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11.5 Power Calculations in Balanced
Three-Phase Circuits
Complex Power in a Balanced Wye Load
Q  V I sin 
QT  3Q  3VL I L sin 
Total reactive power in a balanced three-phase load
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11.5 Power Calculations in Balanced
Three-Phase Circuits
S  P  jQ  V I
*
ST  3S  3VL I L 
Total complex power in a balanced three-phase load
11.5 Power Calculations in Balanced
Three-Phase Circuits
Power Calculations in a Balanced Delta Load
Δ-connected load used to discuss power
calculations
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Note: in a balanced load, av. power per phase is
equal in Y- and Δ-connected.
Reactive power & complex power are also same.
Note:
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Vs =
Check!
Y-to-Y ckt
11.6 Measuring Average Power in
Three-Phase Circuits
A wattmeter measures the average power
delivered to a load by using a current coil
connected in series with the load and a potential
coil connected in parallel with the load.
: stationary & designed to carry a ct. proportional to
the load ct.
: movable & carries a ct. proportional to the load vtg.
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11.6 Measuring Average Power in
Three-Phase Circuits
The key features of the
electrodynamometer wattmeter
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A general circuit whose power
is supplied by n conductors
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11.6 Measuring Average Power in
Three-Phase Circuits
The total average power in a balanced
three-phase circuit can be measured by
summing the readings of two wattmeters
connected in two different phases of the
circuit.
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If we wish to measure the total
power at the terminals of the box,
we need to know n−1 currents and
voltages.
Thus the total power is the sum of
n − 1 product terms; that is
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Only two wattmeters are needed to measure the total
average power in any balanced three-phase system.
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Calculate the reading of each
wattmeter.
Where phase vtg. At the load
= 120 V
a)
b)
c)
d)
Zφ
Zφ
Zφ
Zφ
=
=
=
=
8 + j6Ω
8 - j6Ω
5 + j5√3Ω
10∠-75oΩ
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Home work
Prob. 11.5 11.6 11.11 11.12 11.20 11.21 11.28
11.44 11.46 11.50
11.37
제출기한:
- 다음 요일 수업시간 까지
- 제출기일을 지키지않는 레포트는 사정에서 제외함
EE141
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THE END
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