SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

N
B A B Y
x(t)
(2.6)
The induced voltage is given by the application of Equation (2.5) to Equation (2.6), yielding
Equation (2.7).
dx ( t)
The term
dt
BYx( t)
d d
dx(
t)
g(
t)
B Y
dt dt
dt
(2.7)
is by definition the instantaneous velocity v; thus, Equation (2.7) can be written
as Equation (2.8). This is often called the “BLv law,” where L represents the conductor length;
here Y is used to avoid confusion with inductance).
g( t)
B Y
v
(2.8)
The magnitude of the EMF is directly proportional to conductor speed and is independent of the
presence or value of the resistance shown, although the size of the resistor will determine the
amount of force needed to move the bar. This concept will now be examined.
Lorentz Force Law
The Lorentz Force law is the second of the two fundamental concepts that describe the operation
of electric motors and must be understood in order to create an electrical model of a motor. The
law is generally defined [44] as Equation (2.9),
EvB F q
(2.9)
where F is the force on the particle of charge q moving at velocity v in the electric field of
potential E and magnetic field of flux density B; all bold terms are vector quantities and × is the
vector cross product. In a motor the contribution from E is irrelevant thus Equation (2.9) is
reduced to the more familiar Equation (2.10).
F q v B
(2.10)
When v and B are perpendicular, the vector cross product reduces to a scalar product and yields
Equation (2.11).
F q v B
(2.11)
When Y is perpendicular to B over its entire length and when B is constant over the entire length
Y, Equation (2.11) can be manipulated [68, p.61] to into the equivalent form Equation (2.12),
where Y is the length of the conductor carrying current i; (usually L is used for the length and it is
known as the “BLi law”). It is clear that the force is directly proportional to current.
F B Y
i
(2.12)
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