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5/19/2018 EE 132 - Lecture 1
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EE 132
LECTURE 1
Prepared by:
Engr. Rogelio F. Bersano Jr.
Polyphase Circuits
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Topic Outline
Polyphase Systems
Single-Phase Three-wire Systems
Three Phase Balanced Circuit
The Delta ( ) & Wye (Y) Connection
Power Calculation in Balanced circuit
Wattmeter Method
Three Phase Unbalanced Circuit
Unbalanced Delta and Wye Connected Loads
Power Calculation in Unbalanced circuit
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Course Objective
1. Learn the distinction between single-phase and
polyphase systems.
2. Familiarize with working with both Y- and -
connected three-phase sources and three-phasenetworks.
3. Master the technique of per-phase analysis of
three-phase systems.
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Single Phase System
1. A generator connected through a pair of wire to a
loadSingle Phase Two Wire.
2. Vpis the magnitude of the source voltage, and
is the phase.
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Polyphase System
Circuit or system in which AC sources operate at
the same frequency but different phases are
known as polyphase.
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Polyphase System
Two Phase System:
A generator consists of two coils placed
perpendicular to each other
The voltage generated by one lags the other by90.
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Polyphase System
Three Phase System:
A generator consists of three coils placed 120
apart.
The voltage generated are equal in magnitudebut, out of phase by 120.
Three phase is the most economical polyphase
system.
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Polyphase System
Three Phase System:
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Ideal Y- and- connected voltage
sources
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Real Y- and- connected voltage
sources
Internal impedance of a generator is usually
inductive(due to the use of coils).
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Balanced three-phase source
Three sinusoidal voltages of the same amplitude,
frequency, but differing by 120 phase
difference with one another.
There are two possible sequences:
1. abc (positive) sequence : vb(t) lags va(t) by
120.
2. acb (negative) sequence: vb(t) leads va(t) by120.
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3-phase Advantages
1. The horsepower rating of three-phase motors
and the kVA rating of three-phase transformers
are 150% greater than single-phase motors or
transformers of similar frame size.
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3-phase Advantages
2. The power delivered by a single-phase system
pulsates and falls to zero. The three-phase
power never falls to zero. The power delivered to
the load in a three-phase system is the same atany instant. This produces superior operating
characteristics for three-phase motors.
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3-phase Advantages
3. A three-phase system needs three conductors;
however, each conductor is only 75% the size of
the equivalent kVA rated single-phase
conductors.
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3-phase: Basic Properties
Three-phase systems have either three or four
conductors.
These three-phase conductors are identified as
A, B, and C. The three phases are 120 degrees out of phase
with each other (360 divided by 3).
There is sometimes a fourth conductor, which is
the neutral.
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3-phase: Wye Connection
The wye, or star, connection is made by
connecting one end of each of the phase
windings together in a common node.
Each line terminals has a voltage drop known asthe phase voltage.
The line voltage is measured from phase
conductor to a different phase conductor.
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3-phase: Wye Connection
In a wye system, the line voltage is higher than
the phase voltage by a factor of the square root
of 3 (1.732).
ELine= EPhasex 1.732EPhase= ELine/ 1.732
In a wye system, the line current is equal to the
phase current.ILine= IPhase
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3-phase: Wye Connection
Wye Connected Source
a) abc or positive sequence b) acb or negative sequence
0
120
240
an p
bn p
cn p
V V
V V
V V
0
120
240
an p
bn p
cn p
V V
V V
V V
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3-phase: Wye Connection
3
0 , 120 ,
30
3 90
3 21
120
0
an p bn p cn p
ab an nb an bn
bc bn cn
ca cn an
p
p
an bn p
V
V V V V V V
V V V V V
V V V
V V V
V
V VV
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3-phase: Wye Connection
Line and phase voltages in a wye connection.
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3-phase: Wye Connection
Line and phase currents in a wye connection.
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3-phase: Wye Connection
Vector sum of typical wye system voltages.
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3-phase: Delta Connection
In a delta system, the line current is higher than
the phase current by a factor of the square root
of 3 (1.732).
ILine= IPhasex 1.732
IPhase= ILine/ 1.732
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3-phase: Delta Connection
In a delta system, the line voltage is equal to the
phase voltage.
ELine= EPhase
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3-phase: Delta Connection
Delta Connected Source
a) abc or positive sequence b) acb or negative sequence
Vab
Vbc
Vca
Vab
Vbc
Vca
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3-phase: Delta Connection
3 30
3 30
3 30
a AB CA
b BC AB
c CA BC
AB
BC
CA
I I I
I I I
I
I
I II
I
3
L a b c
p AB BC CA
L p
I I I I
I I I I
I I
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3-phase: Delta Connection
Delta system voltage and current relationships.
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Balanced Three-phase Load
A Balanced load has equal impedances on allthe phases
a) Wye-connected load b) Delta-connected load
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Balanced Three-phase Load
1 2 3
Conversion of Delta circuit to Wye or Wye to Delta.Balanced Impedance Conversion:
Y
a b c
Z Z Z Z
Z Z Z Z
1Z 3 Z
3Y YZ Z
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Balanced Three-phase Network
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Balanced Y-Y Connection
Source
Impedance
Line Impedance
Load
Impedance
Line current Inadd up to zero.
Neutral current is zero:
In= -(
Ia+
Ib+
Ic)= 0
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Balanced Y-Y Connection
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Single Phase Equivalent of Balanced
Y-Y Connection
Balanced three phase circuits can be analyzed on perphasebasis..
We look at one phase, say phase A and analyze the
single phase equivalent circuit.
Because the circuit is balanced, we can easily obtainother phase values using their phase relationships.
ana
Y
VI
Z
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Example 1
Obtain the line currents in the 3-phase circuit of the figureshown below
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Balanced Y-Connection
Three phase sources are usually Wye connected andthree phase loads are Delta connected.
There is no neutral connection for the Y-system.
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Balanced Y-Connection
3
L a b c
p AB BC CA
L p
I I I I
I I I I
I I
Phasor diagram of phase and line currents
Single phase equivalent circuit of the balanced Wye-delta
connection
3
Z
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Example 2
Solve for the line currents in the Y- circuit of the figureshown below. Take Z= 6045
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Balanced-Connection
Both the source and load are Delta connected andbalanced.
, ,BC CAABAB BC CAV VV
I I IZ Z Z
, ,a AB CA b BC AB c CA BC I I I I I I I I I
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Example 3
Refer to the -circuit of the figure shown below. Find theline and phase currents. Assume that the load impedance is
12 + j9 per phase.
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Balanced-Y Connection
30
3
pV
Transforming a Delta connected source
to an equivalent Wye connectionSingle phase equivalent of Delta Wye connection
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Example 4
In the circuit, if Vab = 44010 , Vbc = 440250 , Vca =440130.Find the line currents.
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Power in a Balanced System The total instantaneous power in a balanced three phase system is
constant.
=
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Power in a Balanced System The total instantaneous power in a balanced three phase system is
constant.
=
=
=
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Power in a Balanced System The complex power per phase is Sp. The total complex power for all
phases is S.
p p
p p p p p
2
p p p p
Complex power for each phas
3 cos
1 1= cos = sin
3 3
S V I
3 3 cos 3 cos
3 3 sin 3 sin
S=3S 3V
e
I 3
p p
p p p p p p p
a b c p p p L L
p p p L L
p
p V I
P p V I Q p V I S V I
P jQ
P P P P P V I V I
Q Q V I V I
I Z
p L
2
p
p , , and are
Total
all rm
complex powe
s values, is the load impedance angl
3
3
r
eL
p
L L
V
Z
P jQ
I
V
V I
I
V
S
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Power in a Balanced System2
p2
p p p
p
p
p L, , and are all rms values, is the load impe
To
da
3S=3S 3V I 3
nce
al c
an
3
omplex ower
gle
p
L
p
p
L L
V
VI Z
Z
P
V
V I
I
jQ
I
S
Notice the values of Vp, VL, Ip, ILfor different load connections.
VLVL
VL
Vp Vp
Vp
Ip
Ip Ip
VL
Vp
Ip
VL
VL Vp
Vp
Ip
Ip
Y connected load. connected load.
3L p L pV V I I 3L p L pV V I I
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Example 5
VL=840 V (Rms)
Capacitors for pf
Correction
IL
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Unbalanced Three Phase Systems An unbalanced system is due to unbalanced voltage sources or
unbalanced load.In an unbalanced system the neutral current is NOT zero.
Unbalanced three phase Y connected load.
Line currents DO NOT add up to
zero.
In= -(Ia+ Ib+ Ic) 0
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Example 5
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Power Measurements: The Wattmeter
The wattmeter is a four-terminal device that measures powerdelivered to the network if connected as shown:
the wattmeter
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Measuring Three-Phase Power
The sum of the
powers measured
by the wattmeters
is the total power
delivered.
The common node
xis arbitrary.
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The Two-Wattmeter Method
One wattmeter
can be
eliminated if the
point x ismoved to a line
(as shown, B).
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Example 6
A three-phase system has a line-to-line voltage of 4000 V rms and abalanced Delta-connected load with Z = 40 + j30 ohms. The phase
sequence is abc. Use the two wattmeters connected to lines A and C,
with line B as the common line for the voltage measurement.
Determine the total power measurement recorded by the wattmeters.
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An unbalanced delta-connected load is connected across the lines of a
4-wire balanced three-phase supply with neutral connection. Two
wattmeter W1, W2 are connected in lines 1 and 3 with the common
potential connection in the neutral wire. These wattmeters read 1796.5
and 3000 watts, respectively. The current in lines 1 and 3 is 20 A. If the
common potential connection is transferred to line 2, what will be thereading of the two wattmeters? Solve the power factor of the system.
The system voltage per phase is 208 V. Assume a phase sequence of
1-2-3 with V1nas reference.
Example 7
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END