Download Countertorque and Power of an Electric Generator

p. 63
Countertorque and Power of an Electric Generator
In the previous example we assumed that the generator was operating under a no load condition, that is, that no
current was drawn from the generator by electrical devices attached to it. When an external load it placed across
the generator, current flows through the coil and out of the generator. The presence of current in the coil causes
the magnetic field to exert a torque on the coil (as was discussed previously). By Lenz’s law, this magnetic
torque opposes the torque that turns the generator and so is referred to as a countertorque.
Example
Suppose that the resistance of the coil of the generator of the previous example is 10.0 . A load of 50.0  is
placed across the terminals of the generator. (For simplicity ignore the internal friction of the generator and
assume a purely resistive load.) Assume that enough power is delivered to the generator to maintain an rms
terminal voltage of 120 V.
a. How much current is drawn from the generator?
Since the total resistance of the circuit is 60.0 , I 
V
120 V
; I
; I  2.00 A
R
60.0 
b. What electric power is consumed by the circuit?
P  IV ; P   2.00 A 120 V  ; P  240 W
c. What is the rms countertorque on the coil? Assume three significant figures for all data.
First find the amplitude of the countertorque:
The rms countertorque is then rms


  NBAI ;    4  0.45 T  0.25 m 2  2.00 A 
  0.900 m  N

0.900 m  N

; rms 
; rms  0.636 m  N
2
2
d. What rms mechanical power must be delivered to the generator to maintain its terminal voltage?


P  rms ; P   0.636 m  N  2 60 s-1 ; P  240 W
Note that the mechanical power delivered to the generator matches the power output of the generator. Energy is
conserved. In reality would you expect the mechanical power that must be delivered to the generator to be
greater or smaller than the power output? Why?
p. 64
The Back Emf Generated by an Electric Motor
The heart of any electric motor is a shaft connected to a coil that is suspended in a magnetic field (usually the
magnetic field of a permanent magnet built into the motor). When current passes through the coil, the magnetic
field exerts a torque on the coil, causing it and the shaft to spin. The shaft extends out of the motor where it can
be connected to machinery or some other device that does work. The spinning coil, however, acts as an electric
generator, producing a back emf or counter emf E in the circuit the motor is connected to. If the motor is
connected to a voltage V the current drawn by the motor is then
I
V E
R
where R is the resistance of the motor’s coil.
Example
An electric fan designed to operate off of 120 vac has a coil of 25 turns, cross-sectional area 0.30 m2 and
resistance 100 . The permanent magnet in the fan’s motor produces a magnetic field of 0.50 T. When
operating normally the fan spins at 400 RPM.
a. What is the back emf of the fan while it is operating?


 25 0.50 T  0.30 m 2  400
NBA
E 
; E 
2
2
rev
min
1 min
 2 rad
rev  60 s 
E  110 V
b. What current is drawn by the fan when it is first switched on?
When the fan is first switched on, it is not turning so that the back emf is zero:
I
V E
120 V  0
; I
; I  1.2 A
R
100 
c. What current is drawn by the fan while it is operating?
I
V E
120 V  110 V
; I
; I  0.10 A
R
100 
Note that the fan draws much more current when it is first switched on than when it is operating normally. This
is typical of most appliances, and explains why a circuit breaker on an overloaded circuit is more likely to trip
when a device on that circuit is first switched on.
The current drawn by the fan while it is operating provides the power to balance the friction on the fan’s
moving parts. (Which moving parts experience friction?)