SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

is crucial to the understanding of FOC and is illustrated again in the mathematical derivation and
then again in the development of the space vector.
Mathematical Derivation
The presented graphical depiction of the rotating MMF wave is now developed mathematically.
Equation (3.1) describes each phase’s component MMF as a function of the angle around the
stator periphery. The total MMF as a function of the angle around the stator is given by their sum,
Equation (3.2).
f( , t) fA( ) fB( ) fC(
)
N (3.2)
e
iA( t) cos( ) iB( t) cos( 120 ) iC( t)
cos( 120
)
2
The currents form a balanced sinusoidal set given by Equation (3.3), where Ip is the amplitude
and t e is the electrical position.
i
i
i
A
B
C
I
I
I
p
p
p
cos( t)
cos( t
120
cos( t
120
)
)
Substituting Equation (3.3) into Equation (3.2) yields Equation (3.4).
Ipcos( t) cos( )
N
e
f( , t) Ipcos( t120 )cos( 120
)
2
Ipcos( t120 )cos( 120
)
The amplitude of each cosine term is a function of ωt—that is, each cosine term is a standing
wave in θ with peak that is always located at 0°, 120°, and 240°. Equation (3.4) can be reduced by
the trigonometric identity in Equation (3.5) to give Equation (3.6). Equation (3.6) is the standard
result provided by any machine text (it is valid only for balanced sinusoidal currents).
(3.3)
(3.4)
1
cos( A) cos( B)
cos( A B)
cos( A B)
(3.5)
2
3 N e
( , t)
I
cos( t
)
2 2
f p
It can be noticed that the order in which (A and B) in Equation (3.5) are assigned to (ωt and θ) is
arbitrary, thus Equation (3.6) could also be written as Equation (3.7). This also follows from the
fact that cosine is an odd function; both equations are found in the literature.
3 Ne
f ( , t) Ipcos( t)
2
2
(3.6)
(3.7)
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