CHAPTER 4: THREEPHASE CIRCUIT

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CHAPTER 4 THREE-PHASE CIRCUIT

• Balanced Three-Phase Voltages
• Balance Connections Wye-Wye, Wye-Delta,
  Delta-Delta, Delta-Wye
• Power in a Balanced System

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Introduction
Single phase systems
a) two-wire type b) three-wire type
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Introduction
Three Phase System
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Balanced Three Phase Voltages
Three-phase voltage sources
a) wye-connected source b) delta-connected
source
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Balanced Three Phase Voltages
Phase sequences
a) abc or positive sequence b) acb
or negative sequence
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Balanced Three Phase Voltages
If the voltage source have the same amplitude and
frequency ? and are out of phase with each other
by 120o, the voltage are said to be balanced.
Balanced phase voltages are equal in magnitude
and out of phase with each other by 120o
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Balanced Three Phase Voltages
abc sequence or positive sequence
is the effective or rms value
acb sequence or negative sequence
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Balanced Three Phase Voltages
Two possible three-phase load configurations
a) a wye-connected load b) a
delta-connected load
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Balanced Three Phase Voltages
A balanced load is one in which the phase
impedances are equal in magnitude and in phase.
For a balanced wye connected load
For a balanced delta connected load
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Example 1
Determine the phase sequence of the set of
voltages van 200 cos(?t 10?) vbn 200 cos(?t
- 230?), vcn 200 cos(?t - 110?) 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.

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Exercise 1

Given that , find
Van and Vcn, assuming a positive (abc)
sequence. Answer
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Balanced Wye-Wye Connection
A balanced Y-Y system is a three phase system
with a balanced Y connected source and balanced Y
connected load.
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Balanced Wye-Wye Connection
Source impedance
Line impedance
Load impedance
Total impedance per phase
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Balanced Wye-Wye Connection
Line to line voltages or line voltages
Magnitude of line voltages
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Balanced Wye-Wye Connection
Phasor diagram illustrating the relationship
between line voltages and phase voltages.
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Balanced Wye-Wye Connection
Balanced Y-Y connection
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Balanced Wye-Wye Connection
Applying KVL to each phase
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Balanced Wye-Wye Connection
A single phase equivalent circuit
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Example 2
Calculate the line currents in the three wire Y-Y
system of figure below.
Reference Alexander, Sadiku Chapter 12 - page
511
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Exercise 2
A Y-connected balanced three-phase generator with
an impedance of 0.4j0.3 O per phase is connected
to a Y-connected balanced load with an impedance
of 24 j19 O per phase. The line
joining the generator and the load has an
impedance of 0.6 j0.7 O per phase. Assuming a
positive sequence for the source voltages and
that Find (a) the line voltages (b)
the line currents
Reference Alexander, Sadiku Chapter 12 PP 12.2
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Balanced Wye-Delta Connection
A balanced Y- ? system consists of balanced Y
connected source feeding a balanced ? connected
load.
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Balanced Wye-Delta Connection
Line voltages
Phase currents
Line currents
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Balanced Wye-Delta Connection
A single-phase equivalent circuit of a balanced
Y-? circuit
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Balanced Wye-Delta Connection
From phasor diagram
Magnitude line currents
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Balanced Wye-Delta Connection
A single phase equivalent circuit
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Example 3
A balanced abc sequence Y-connected source
with Is connected to a ?-connected balanced load
(8j4)O per phase. Calculate the phase and line
currents.
Reference Alexander, Sadiku Chapter 12 - page
513
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Exercise 3
One line voltage of a balanced Y-connected
source is If the source is connected to a ?
-connected load of , find the
phase and line currents. Assume the abc sequence.
Reference Alexander, Sadiku Chapter 12 PP 12.3
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Balanced Delta-Delta Connection
A balanced ? - ? system is one in which both
balanced source and balanced load are ? connected.
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Balanced Delta-Delta Connection
Phase currents
Line voltages
Line currents
Magnitude line currents
Total impedance
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Example 4
A balanced ? connected load having an impedance
20-j15 O is connected to a ? connected, positive
sequence generator having Calculate the phase
currents of the load and the line currents.
Reference Alexander, Sadiku Chapter 12 - page
515
31
Exercise 4
A positive-sequence, balanced -connected source
supplies a balanced ?-connected load. If the
impedance per phase of the load is 18j12 O
and , find IAB and
VAB.
Reference Alexander, Sadiku Chapter 12 PP 12.4
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Balanced Delta-Wye Connection
A balanced ? -Y system consists of balanced ?
connected source feeding a balanced Y connected
load.
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Balanced Delta-Wye Connection
Applying KVL to loop aANBba
From
Line currents
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Balanced Delta-Wye Connection
Replace ? connected source to equivalent Y
connected source.
Phase voltages
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Balanced Delta-Wye Connection
A single phase equivalent circuit
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Example 5
A balanced Y connected load with a phase
resistance of 40 O and a reactance of 25 O is
supplies by a balanced, positive sequence ?
connected source with a line voltage of 210 V.
Calculate the phase currents. Use Vab as
reference.
Reference Alexander, Sadiku Chapter 12 - page
519
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Exercise 5
In a balanced -Y circuit, and ZY (12 j15)
O. Calculate the line currents.
Reference Alexander, Sadiku Chapter 12 PP 12.5
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POWER IN A BALANCED SYSTEM
For Y connected load, the phase voltage
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POWER IN A BALANCED SYSTEM
Phase current lag phase voltage by ?.
If
The phase current
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POWER IN A BALANCED SYSTEM
Total instantaneous power
Average power per phase
Reactive power per phase
Complex power per phase
Apparent power per phase
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POWER IN A BALANCED SYSTEM
Total average power
Total reactive power
Total complex power
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POWER IN A BALANCED SYSTEM
Power loss in two wires
Power loss in three wires
PL power absorbed by the load IL magnitude of
line current VL line voltage R line
resistance
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Example 6
A three-phase motor can be regarded as a balanced
Y-load. A three-phase motor draws 5.6 kW when
the line voltage is 220 V and the line current
is 18.2 A. Determine the power factor of the
motor.
Reference Alexander, Sadiku Chapter 12 - page
523
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Exercise 6
Calculate the line current required for a 30-kW
three-phase motor having a power factor of 0.85
lagging if it is connected to a balanced source
with a line voltage of 440 V.
Reference Alexander, Sadiku Chapter 12 PP 12.7
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Exercise 7
For the Y-Y circuit in Exercise 2, calculate the
complex power at the source and at the load.
Reference Alexander, Sadiku Chapter 12 PP 12.6