ee132 lec 1 polyphase circuits
DESCRIPTION
Circuit AnalysisTRANSCRIPT
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EE 132EE 132Electric Circuit Electric Circuit Theory IITheory II
Lecture 1
Polyphase Circuits
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Review: 1-Review: 1-φφ circuits circuitsReview: 1-Review: 1-φφ circuits circuits
Two-wire type
Three-wire type
VVp p = magnitude of the source voltage
φφ = phase of source voltage
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Polyphase CircuitsPolyphase CircuitsPolyphase CircuitsPolyphase Circuits
Circuits or systems in which the ac sources operate at the same frequency but different phases
Two-phase, three-wire system
The source is a generator with two coils placed in perpendicular to each other so that the voltage generated by one lags the other by 90°.
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Polyphase CircuitsPolyphase CircuitsPolyphase CircuitsPolyphase Circuits
Circuits or systems in which the ac sources operate at the same frequency but different phases
Three-phase, four-wire system
The source is 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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Why 3-Why 3-φφ systems? systems?Why 3-Why 3-φφ systems? systems? All electric power is generated and distributed in 3-φ, at
the operating frequency of 60 Hz (or ω = 377 rad/s) or 50 Hz (or ω = 314 rad/s)
The instantaneous power can be constant (not pulsating) Uniform power transmission & less vibration of 3-φ
machines
For the same amount of power, 3-φ is more economical Less volume of wire needed
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Balanced 3-Balanced 3-φφ sources sourcesBalanced 3-Balanced 3-φφ sources sources
Three voltages sources connected to loads by 3 or 4 wires
Equivalent to three (3) single phase circuits
Can be connected in WYE (Y) or DELTA (Δ)
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Balanced 3-Balanced 3-φφ sources sources
Balanced phase voltagesBalanced phase voltages are equal in magnitude and are out of phase by 120°.
Y-connected source Δ-connected source
Vp = phase voltageVL = line voltage
0an bn cn
an bn cn
V V V
V V V
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Balanced 3-Balanced 3-φφ sources sources
The phase sequencephase sequence is the time order in which the phase voltages reach their peak values wrt time.
abc (positive) Phase sequence
acb (negative) Phase sequence
0
120
240 120
an p
bn p
cn p p
V
V
V V
V
V
V
0
120
240 120
an p
bn p
cn p p
V
V
V V
V
V
V
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Balanced 3-Balanced 3-φφ load load
A balanced loadbalanced load is one in which the phase impedances are equal in magnitude and in phase. Otherwise, load is unbalanced.
Y-connected load Δ-connected load
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Balanced 3-Balanced 3-φφ load load
Y-connected load
1 2 3 Y Z Z Z Z
A balanced loadbalanced load is one in which the phase impedances are equal in magnitude and in phase. Otherwise, load is unbalanced.
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Balanced 3-Balanced 3-φφ load load
Δ-connected load
A B C Z Z Z Z
A balanced loadbalanced load is one in which the phase impedances are equal in magnitude and in phase. Otherwise, load is unbalanced.
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Balanced 3-Balanced 3-φφ load load
Y-connected load Δ-connected load
or3 Y Z Z 1
3Y Z Z
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Balanced 3-Balanced 3-φφ systems systems
We can have four possible combinations:
Y-Y connection (Y-connected source, Y-connected load)
Y-Δ connection Easy to remove and add loads connected in delta
Δ -Δ connection
Δ-Y connectionNot common because of the circulating current that will result in the delta windings of the source if the phase voltages are slightly unbalanced
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source impedance
line impedance
load impedance
neutral impedance
Balanced Y-Y connectionBalanced Y-Y connectionA balanced Y-Y systembalanced Y-Y system is a 3-φ system with a balanced
Y-connected source & a balanced Y-connected load.
total load impedance per phaseY
S l L
Z
Z Z Z
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Balanced Y-Y connectionBalanced Y-Y connectionA balanced Y-Y systembalanced Y-Y system is a 3-φ system with a balanced
Y-connected source & a balanced Y-connected load.
= total load impedance per phase
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Balanced Y-Y connectionBalanced Y-Y connection
3 30
3 90
3 210
ab p
bc p
ca p
V
V
V
V
V
V
Assuming positive phase sequence:
The phase voltages are
The line voltages are
0
120
120
an p
bn p
cn p
V
V
V
V
V
V
0 120
1 31 3 30
2 2
3 90
3 210
ab an nb an bn p p
p p
bc bn cn p
ca cn an p
V V
V j V
V
V
V V V V V
V V V
V V V
3L pVV
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Balanced Y-Y connectionBalanced Y-Y connection
Where
and
3L pVV
p an bn cnV V V V
L ab bc caV V V V
3 30
3 90
3 210
ab p
bc p
ca p
V
V
V
V
V
V
0
120
120
an p
bn p
cn p
V
V
V
V
V
V
0a b c I I I
0n a b c I I I I
0nN n n V Z I L PI I
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Balanced Y-Y connectionBalanced Y-Y connection
Where
And
Define:
For the Y-Y connection:
3L pVV
p an bn cnV V V V
L ab bc caV V V V
3 30
3 90
3 210
ab p
bc p
ca p
V
V
V
V
V
V
0
120
120
an p
bn p
cn p
V
V
V
V
V
V
L PI I
IIPP = phase current
= current in each phase of the source/load
IILL = line current
= current in each line
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Examples:1. Calculate the line currents in
the circuit shown.
Balanced Y-Y connectionBalanced Y-Y connection
6.81 21.8
6.81 141.8
6.81 98.2
a
b
c
I
I
I
2. A Y-connected balanced three-phase generator with an impedance of 0.4 + j0.3 Ω per phase is connected to a Y-connected balanced load with an impedance of 24 + j19 Ω. The line joining the generator and the load has an impedance of 0.6 + j0.7 Ω per phase. Assuming a positive sequence for the source voltages and that Van = 120∟30° V, find: (a) the line voltages; (b) the line currents.
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Balanced Y-Balanced Y-ΔΔ connection connectionA balanced Y-balanced Y-ΔΔ system system consists of a balanced Y-
connected source feeding a balanced Δ-connected load.
3 30
3 90
3 210
ab p
bc p
ca p
V
V
V
V
V
V
0
120
120
an p
bn p
cn p
V
V
V
V
V
V
The phase currents are:
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Balanced Y-Balanced Y-ΔΔ connection connectionA balanced Y-balanced Y-ΔΔ system system consists of a balanced Y-
connected source feeding a balanced Δ-connected load.
3 30
3 90
3 210
ab p
bc p
ca p
V
V
V
V
V
V
0
120
120
an p
bn p
cn p
V
V
V
V
V
V
To get the line currents, apply KCL at nodes A, B & C:
a AB CA
b BC AB
c CA BC
I I I
I I I
I I I 3L PI I
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Balanced Y-Balanced Y-ΔΔ connection connectionA balanced Y-balanced Y-ΔΔ system system consists of a balanced Y-
connected source feeding a balanced Δ-connected load.
3 30
3 90
3 210
ab p
bc p
ca p
V
V
V
V
V
V
0
120
120
an p
bn p
cn p
V
V
V
V
V
V
To get the line currents, apply KCL at nodes A, B & C:
a AB CA
b BC CA
c CA BC
I I I
I I I
I I I 3L PI I
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Example:
Balanced Y-Balanced Y-ΔΔ connection connection
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Balanced Y-Balanced Y-ΔΔ connection connectionExample:
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Example:
Balanced Y-Balanced Y-ΔΔ connection connection
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Balanced Balanced ΔΔ - -ΔΔ connection connectionA balanced balanced ΔΔ - -ΔΔ system system is one in which both the
balanced source and balanced load are Δ-connected.
0
120
120
ab p
bc p
ca p
V
V
V
V
V
V
The phase currents are:
Assuming no line impedances,
ab AB
bc BC
ca CA
V V
V V
V V
The line currents are:
3L PI I
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Example:
Balanced Balanced ΔΔ - -ΔΔ connection connection
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Example:
Balanced Balanced ΔΔ - -ΔΔ connection connection
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Example:
Balanced Balanced ΔΔ - -ΔΔ connection connection
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Balanced Balanced ΔΔ -Y connection -Y connectionA balanced balanced ΔΔ - -YY system system consists of a balanced Δ -
connected source feeding a balanced Y-connected load.
0
120
120
ab p
bc p
ca p
V
V
V
V
V
V
These are also the line voltages.
To obtain the line currents, we can apply KVL to loop aANBba i.e
0
0
0
ab Y a Y b
Y a b ab p
pa b
Y
V
V
V Z I Z I
Z I I V
I IZ
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Balanced Balanced ΔΔ -Y connection -Y connectionA balanced balanced ΔΔ - -YY system system consists of a balanced Δ -
connected source feeding a balanced Y-connected load.
0
120
120
ab p
bc p
ca p
V
V
V
V
V
V
These are also the line voltages.
To obtain the line currents, we can apply KVL to loop aANBba i.e
The line currents are:
0
0
0
ab Y a Y b
Y a b ab p
pa b
Y
V
V
V Z I Z I
Z I I V
I IZ
But for the abc phase sequence,Thus
120 ,b a I I
1 1 120 ,
3 30
a b a
a
I I I
I
3 30pa
Y
V I
Z
120
120
b a
c a
I I
I I
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Balanced Balanced ΔΔ -Y connection -Y connectionA balanced balanced ΔΔ - -YY system system consists of a balanced Δ -
connected source feeding a balanced Y-connected load.
0
120
120
ab p
bc p
ca p
V
V
V
V
V
V
These are also the line voltages.
To obtain the line currents, we can apply KVL to loop aANBba i.e
0
0
0
ab Y a Y b
Y a b ab p
pa b
Y
V
V
V Z I Z I
Z I I V
I IZ
L PI I
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Balanced Balanced ΔΔ -Y connection -Y connectionA balanced balanced ΔΔ - -YY system system consists of a balanced Δ -
connected source feeding a balanced Y-connected load.
Alternatively, to obtain the line currents, we can also transform the ΔΔ-connected load into a Y-connected load.
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SummarySummary
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SummarySummary
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Example:
Balanced Balanced ΔΔ -Y connection -Y connection
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Example:
Balanced Balanced ΔΔ -Y connection -Y connection
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Example:
Balanced Balanced ΔΔ -Y connection -Y connection