SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Elementary Physics and Modeling
This section examines the fundamental physics of electromechanical energy conversion. The laws
of Faraday and Lentz are developed. The basic magnetic structure of a brushless motor is
introduced to derive the electrical model and discus flux linkage.
Faraday’s Law
Faraday’s law of induction describes the voltage induced in a coil of wire in relation to the sum of
flux linked by that coil. This is the first of two fundamental concepts that describe the operation
of electric motors and must be understood to create an electrical model of a motor. Physics texts
often define this law as Equation (2.1), where φ represents the flux through a loop of wire in a
plane perpendicular to the flux. 2
d
g( t)
(2.1)
dt
This idealized situation is not practical when the coil consists of multiple turns which are
potentially displaced in space, in which case the flux through each turn may not be the same for
every turn. To account for this condition, φ must be replaced with something that accounts for the
additional turns of the coil. Namely, it must be replaced with the total flux linkage given by
Equation (2.2), where φn is the flux that links the n th turn and N is the total number of turns. This
can be interpreted as considering each turn of the coil to be a separate, single-turn coil connected
in series with the other turn.
N
n 1
n
For a concentrated winding (where all the turns are grouped together in two slots 180° electrical
degrees apart) the assumption can be made that the same flux links each of the N turns of the coil.
With this assumption, Equation (2.2) is reduced to Equation (2.3). If φ in Equation (2.1) is
replaced with λ in Equation (2.3), Equation (2.1) becomes Equation (2.4). Later on, λ will be
identified as the coil’s “flux linkage.”
(2.3)
(2.2)
d
d
g(
t)
N
(2.4)
dt dt
2 Generally this induced voltage is denoted by e(t) or ε(t) but this report will use g(t).
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