EE 132EE 132Electric Circuit Electric Circuit Theory IITheory II
Lecture 1
Polyphase Circuits
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
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°.
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°.
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
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)
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
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
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
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.
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.
Balanced 3-Balanced 3-φφ load load
Y-connected load Δ-connected load
or3 Y Z Z 1
3Y Z Z
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
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
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
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
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
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
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.
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:
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
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
Example:
Balanced Y-Balanced Y-ΔΔ connection connection
Balanced Y-Balanced Y-ΔΔ connection connectionExample:
Example:
Balanced Y-Balanced Y-ΔΔ connection connection
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
Example:
Balanced Balanced ΔΔ - -ΔΔ connection connection
Example:
Balanced Balanced ΔΔ - -ΔΔ connection connection
Example:
Balanced Balanced ΔΔ - -ΔΔ connection connection
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
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
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
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.
SummarySummary
SummarySummary
Example:
Balanced Balanced ΔΔ -Y connection -Y connection
Example:
Balanced Balanced ΔΔ -Y connection -Y connection
Example:
Balanced Balanced ΔΔ -Y connection -Y connection