Environmental Engineering Reference
In-Depth Information
subject to a restraining force F ,
=
F
IBL
(4.1)
In addition, if the wire moves at a velocity V in the direction of the force, it experiences an electric
field E in the direction opposite to that of the current in the amount 6
E
=
VB
(4.2)
In a generator or motor, wires attached to a rotating armature move through a magnetic field
established by a field coil or permanent magnet. The force F applied to a generator armature wire
delivers mechanical power
at a rate FV while the electrical power produced when the current I
flows in the direction of the potential increase EL is IEL . Utilizing equations (4.1) and (4.2), these
powers are found to be equal: 7
P
P =
FV
=
IBLV
=
IEL
(4.3)
Neglecting any electrical and mechanical losses, the mechanical power input ( FV ) to an ideal
generator then equals the electrical power output ( IEL ).
If we reverse the direction of the velocity V , the direction of the mechanical power is also
reversed; that is, there is a mechanical power output from the device. Simultaneously, the electric
field is reversed and electric power is now an input. In this mode the device is an electric motor.
Neglecting losses, equation (4.3) defines the equality of input electrical power to output mechanical
power for an electric motor.
Figure 4.3(b) shows how a rectangular loop of wire attached to a rotating armature and con-
nected to slip rings can deliver the current to an external stationary circuit via brushes that contact
the rotating slip rings. For this geometry, the peripheral velocity V of the armature wires is 2
rf ,
where f is the armature's rotational frequency and r is the distance of the wire from the armature
axis, so that the ideal output and input power
π
P
is, using equation (4.3),
P =
2 IBL
(
2
π
rf
) =
I
(
4
π
frLB
)
(4.4)
where the factor of 2 arises from the return leg of the circuit. Because the electrical power equals
the product of the current I times the potential difference
φ
across the electrical output terminals,
the factor
. 8
In the case of the armature circuit of Figure 4.3(b), the electric potential changes algebraic sign
as the armature loop rotates through 180 degrees, thus producing alternating current (AC) power
(
4
π
frLB
)
in equation (4.4) is equal to
φ
6 The general forms of equations (4.1) and (4.2), using boldface characters to represent vector quantities, are,
respectively, F
B .
7 Here we are ignoring any resistive loss in the wire.
8 The potential increase φ of an electric generator is called an electromotive force . The generator's internal
current I flows in the direction of increasing electric potential φ , whereas the current in an external circuit
connected to the generator flows in the direction of decreasing potential. A source of electromotive force,
such as a generator or battery, is a source of electric power for the attached external circuit.
=− (
I
×
B
)
L and E
=−
V
×
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