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Circuits II
EE221
Unit 12
Instructor: Kevin D. Donohue
Three Phase Circuits, Balanced Y-Y, Y-, and
- Three-Phase Circuits
Polyphase Circuits
Polyphase circuits contain multiple sources at the same
frequency but different phases. Power is distributed
over the power grid in the form of three-phase
sinusoids.
Advantages of Three-Phase power distribution include:
 (Constant Power) Instantaneous power can be constant in a
three phase system.
 (More Economical) For equivalent power, the 3-Phase systems
are more economical than single-phase (can be driven with
lower currents and voltages, and fewer wires required because
of a common neutral connection between the phases).
 (Flexible) Single phase service can be extracted from the 3phase systems or phases manipulated to create additional
phases.
Balanced 3-Phase Voltages
Balanced phase voltage are equal in magnitude and separate by 120
degrees in phase.
Voltages generated from a 3-phase generator can have 2 phase
sequence possibilities depending on direction of the rotor:
Positive sequence (Counter Clockwise Rotation):
Vˆan  V p 0
Vˆbn  V p   120
Vˆcn  V p   240  V p 120
Negative sequence (Clockwise Rotation):
Vˆan  V p 0
Vˆbn  V p   240  V p 120
Vˆcn  V p   120
Show that the sum of all phase voltages in a balanced system is zero.
Single and 3-Phase Circuit Comparison
Consider the phase voltages of equal amplitude
Vˆp  Vˆan  Vˆbn  Vˆcn
Show that the line voltages are given by:
Vˆab  Vˆac  Vˆbc  3 Vˆp
In general:
Vˆab  3V p 30
Vˆbc  3V p   90
Vˆca  3V p   210
0º
Balanced 3-Phase Voltage Connections
There are 2 ways to connect a Balanced set of sources:
Y (wye)-Connected
 (delta)-Connected
Balanced Loads
Balanced loads are equal in magnitude and phase.
There 2 ways to connect balanced loads
Y (wye)-Connected
A
B
N
 (delta)-Connected
C
A
B
C
Show that for equivalent loads Z = 3ZY
Load-Source Connections
There are 4 possible ways balanced sources and loads can be
connected:
Y
Source to Y Load (Y-Y)
  Source to  Load (-)
 Y Source to  Load (Y-)
  Source to Y Load (-Y)
If not specified, the voltages on the sources will be assumed
to be in RMS values.
Balanced Y-Y Connection
The complete Y-Y connection is shown below with impedances listed
separately for the source (subscript s), line (subscript l), and load
(subscript L).
V p  Vˆan  Vˆan  Vˆan
VL  Vˆab  Vˆbc  Vˆca
For a positive sequence with
Vˆ an V p 0, it can be shown that
Vˆab  3V p 30
Vˆbc  3V p   90
Vˆca  3V p   210
Balanced Y-Y Connection
Show that the current in each phase can be expressed as:
ˆ
ˆI  Van ,
Iˆb  Iˆa   120,
Iˆc  Iˆa   240 ,
a
ZY
and that Iˆa  Iˆb  Iˆc  Iˆn  0
Iˆa
Ẑ Y
Iˆn
Ẑ Y
Iˆc
Ẑ Y
Iˆb
Because of the symmetry
of a balanced 3 phase
system, the neutral
connection can be dropped
and the system analyzed
on a per phase basis. In a
Y-Y connected system, the
phase (source or load) and
line currents are the same.
Balanced Y- Connection
In this case the line voltages are directly across each load. It can be shown
that:
3Vˆp Vˆab VˆAB
ˆ
ˆ
ˆ
ˆ
ˆ
I AB 
Z

Z

Z
,
I BC  I AB  120,
I CA  I AB  240
and the load currents and phase currents are related by:
Iˆa  IˆAB 3 - 30
Note the –connected load
can be converted to a Yconnected load through:
ˆ
Z
ZˆY  
3
Balanced  - Connection
In this case the line voltages are the phase voltages and are
directly across each load. It can be shown that:
ˆ
ˆ
ˆI  Vab  VAB ,
AB
Z Z
IˆBC  IˆAB  120,
IˆCA  IˆAB  240
The line currents can be obtained from the phase currents
Iˆ  Iˆ 3 - 30
a
AB
Balanced  -Y Connection
In this case the phase voltages are across the lines. It can be shown that:
Vˆab  V p 0,
Vˆbc  V p   120,
Vˆca  V p 120
the line current is related to the phase voltage by:
V
Iˆa  ab  - 30
3ZY
Note the –connected
source can be converted
to a Y-connected source
through:
Vˆab
ˆ
Van 
  30
3
Power in Balanced System
Show that the instantaneous power absorbed by a load in a balanced
Y-Y system is a constant given by:
p(t )  3V p I p cos( )
where the impedance in a single phase is given by:
ZˆY  Z
The complex power per phase is
S  V p I p exp( j )
Note that average power or real power is the same as the
instantaneous power for the 3-phase system.