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Chapter 12
Polyphase Circuits
1
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Polyphase (in practice,
three-phase) voltage
sources are important
for transmission over
the grid, and for large
industrial loads.
These three sources are 120°
out of phase with each other.
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

The notation Vab indicates
the voltage between point a
(+) and point b (−).
Indicating the + and −
terminals is redundant if
double-subscript voltages
are used.
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KVL equations can be written correctly without
reference to a circuit diagram:
Vad  Vab Vbd
Vad  Vab Vbc Vcd
Vab  Vba
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Vab  Van Vnb
 Van Vbn
This is an example three-phase
source with a neutral.
 173.2 30 o
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The double subscript notation Iab means the
current from a to b by the direct path
When the direct path
is not obvious
(e.g. between c and d)
we need to use a different
notation.
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The typical North American household is
provided single-phase 3-wire power, where
V1≅110 V and Vab=2V1.
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When the A and B loads are
balanced, the neutral wire
from n to N carries no
current:
Van
Vnb
I aA 
 I Bb 
Zp
Zp
InN  I Bb  I aA  I Bb  I Aa  0
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Determine the power delivered to each of the
three loads as well as the power lost in the
neutral wire and each of the two lines.
Answer:
P50 = 206 W
P100 =117 W
P20+j10 = 1763 W
PaA = 126 W
PbB = 108 W
PnN =3 W
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A balanced three phase source has
Van  Vbn  Vcn
and


Van Vbn Vcn  0
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The abc (positive) and cba (negative) sequences:
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The line voltages have amplitude
VL  3Vp

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For a balanced source and balanced loads, the
neutral wire current INn=0 and so the neutral
wire could have any impedance, including ∞.
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Find the total power delivered to the loads.
Answer: P=600 W
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A balanced three-phase system with a line
voltage of 300 V is supplying a balanced Yconnected load with 1200 W at a leading PF of
0.8.
Find the line current and the per-phase load
impedance.
Answer: IL = 2.89 A,
Zp = 60ej36.9◦ Ω
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A delta-connected load is also commonly used
(note the absence of the neutral wire).
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An example phasor
diagram for an
inductive phase
impedance.
Note that
IL  3Ip
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The wattmeter is a four-terminal device that
measures power delivered to the network if
connected as shown:
the wattmeter
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The sum of the
powers measured
by the wattmeters
is the total power
delivered.
The common node x
is arbitrary.
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One wattmeter
can be
eliminated if
the point x is
moved to a line
(as shown, B).
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