Transcript Per Phase Analysis

Lesson 35
Per Phase Analysis
Learning Objectives
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Derive the relationship between line to line voltages and line
to neutral voltages in a balanced Y-Y three phase circuit.
For a balanced Y-Y three phase circuit convert the line to
line voltage phasor to the line to neutral voltage phasor and
vice versa.
Plot the phasors of line to line voltages and line to neutral
voltages for a Y-Y three phase circuit.
Derive the relationship between line current and phase load
current in a balanced Y- Δ three phase circuit.
For a balanced Y- Δ three phase circuit convert the line
current phasor to phase load current phasor and vice versa.
Plot the phasors of line current and phase load current in a
balanced Y- Δ three phase circuit.
Review
Y-Y system
Y Load
Generator
Lines
PHASE/LINE VOLTAGES
Line and phase voltages of the
Y-connected three-phase
generator.
Determining a line voltage for
a three-phase generator.
Line/Phase Voltages for a Wye Circuit
Using the following relationship
Vab  3Van30
we can also convert back…
Vab
Van 
330
E AN
VAB

330
Therefore, given any voltage at a
point in a balanced, 3-phase Y
system, you can determine the
remaining 5 voltages by inspection.
Law of Sines
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The ratio of the length of a side to the sine of its
corresponding opposite angle is constant
a
b
c
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sin  sin  sin 
 Applying the law of sines to the vector EAB:
  120
180  120

 30
2
E AN
E AB

sin 30 sin120
sin120
E AB  E AN
 E AN 3
sin 30
EAN
EBN
Nominal voltages
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If phase voltage Van = 120 V then line
voltages Vab = 208 V.
This is referred to as 120/208-V system.
Two other common nominal voltages are
220/380-V and 347/600-V.
Currents for a Wye Circuit
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For Y loads, line current and phase current are the
same.
I a  Van / Z an
Line current
Phase current
Wye-Wye Single Phase Equivalent
Original circuit
Single-phase equivalent
Example Problem 1
EAB = 2080 V .
Find the phase voltages and line currents.
 load definitions
Review
Y-Y
For  loads, phase voltage and line
voltage are the same thing.
Line currents are the currents in the line conductors.
Phase currents are the currents through phases .
Line/Phase Currents for a Delta Circuit
Ibc
Line current
Phase current
Line/Phase Currents for a Delta Circuit
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Relationship between line and phase currents
I ab  Vab / Z ab
I a  I ab  I ca
I a  3I ab  30
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For a balanced  system, the magnitude of line current
is 1.732 times the magnitude of the phase current and
line current lags phase current by 30.
Therefore, given any current at a point in a balanced,
3-phase  system, you can determine the remaining 5
currents by inspection.
Line/Phase Currents for a Delta Circuit
Phasor diagram
Equivalent Ckt to solve Wye-Delta Problem
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This is just an equivalent circuit of the load side!
What is missing from the equivalent circuit?
 Ia , Eab , Ean
You can only use the equivalent circuit to solve for load side voltages
and currents:
 Vab, Iab
Notice that this equivalent circuit uses LINE VOLTAGE

#1 mistake with DELTA circuits is using the wrong kind of voltage!
Example Problem 2
Ia = 41.66.9 A .
Determine the phase currents in the load.
Determine the supply line voltage EAB and the supply phase
voltage EAN.
Line Impedances in 3 Phase AC
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Impedances in transmission lines complicate
three phase analysis
Voltage drop in the transmission line must be
accounted for: EAN will not be the same as Van
VAa
Wye-Wye Single Phase Equivalent
Original circuit
Single-phase equivalent
•Notice that the neutral line impedance drop is ignored on the
single phase equivalent.
•No voltage drop occurs across this neutral line since in
reality, no current flows through it.
Wye-Wye Single Phase Equivalent
Original circuit
+ VAa -
Single-phase equivalent
•Use KVL on the single phase equivalent to find unknowns:
EAN –VAa –Van = 0
Example Problem 3
EAB = 2080 V. Determine:
a. Line Currents
b. Load phase voltages
Delta To Wye (∆→Y) Conversion
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A Delta circuit with line impedances cannot be
solved without first performing a ∆→Y conversion.
The Delta load can be converted to an equivalent Wye
load
Z
ZY 
3
Equivalent Delta Ckt with LINE IMPEDANCES
WYE to DELTA CONVERSION
ZY 
Z  15  21 j

 57 j
3
3
Example Problem 4
EAB = 2080 V. Determine:
a. Find load phase voltage Vab
b. Find the load phase current Iab .