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103
Chapter 4
:
AC Machinery Fundamentals
41.
The simple loop is rotating in a uniform magnetic field shown in Figure 41 has the following
characteristics:
B
=
05
. T to the right
r
=
01
. m
l
=
05
. m
ω
= 103 rad/s
(a)
Calculate the voltage
e
t
tot
( ) induced in this rotating loop.
(b)
Suppose that a 5
Ω
resistor is connected as a load across the terminals of the loop.
Calculate the
current that would flow through the resistor.
(c)
Calculate the magnitude and direction of the induced torque on the loop for the conditions in
(b)
.
(d)
Calculate the electric power being generated by the loop for the conditions in
(b)
.
(e)
Calculate the mechanical power being consumed by the loop for the conditions in
(b)
.
How does
this number compare to the amount of electric power being generated by the loop?
ω
m
r
v
ab
v
cd
B
N
S
B
is a uniform magnetic
field, aligned as shown.
a
b
c
d
S
OLUTION
(a)
The induced voltage on a simple rotating loop is given by
( )
ind
2
sin
e
t
r Bl
t
=
ω
(48)
( ) ( )( )( )( )
ind
2 0.1 m 103 rad/s 0.5 T 0.5 m sin103
e
t
t
=
( )
ind
5.15 sin103 V
e
t
t
=
(b)
If a 5
Ω
resistor is connected as a load across the terminals of the loop, the current flow would be:
( )
ind
5.15 sin 103 V
1.03 sin 103 A
5
e
t
i t
t
R
=
=
=
Ω
(c)
The induced torque would be:
( )
ind
2
sin
t
ril
g506
τ
θ
=
(417)
( ) ( )( )( )( )
ind
2 0.1 m 1.03 sin
A 0.5 m 0.5 T sin
t
t
t
=
( )
2
ind
0.0515 sin
N m, counterclockwise
t
t
=
⋅
(d)
The instantaneous power generated by the loop is:
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( ) ( )( )
2
ind
5.15 sin
V 1.03 sin
A
5.30 sin
W
P t
e i
t
t
t
ω
=
=
=
The average power generated by the loop is
2
ave
1
5.30 sin
2.65 W
T
P
t dt
T
=
=
(e)
The mechanical power being consumed by the loop is:
( )( )
2
2
ind
0.0515 sin
V 103 rad/s
5.30 sin
W
P
t
t
τ
=
=
=
Note that the amount of mechanical power consumed by the loop is equal to the amount of electrical power
created by the loop.
This machine is acting as a generator, converting mechanical power into electrical
power.
42.
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This note was uploaded on 10/15/2009 for the course EE 255 taught by Professor Shaban during the Spring '05 term at Cal Poly.
 Spring '05
 Shaban
 Volt

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