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Then the value of α that minimizes this objective function is the optimal
commutation angle. It can be easily shown that the minimum occurs at α =
π/6and,
³
´
3π+4π 2 )
π
6
= π(72−72
3+3
minO(α)=O
.
(4.99)
108
In other words, while using CC-BLDC mode, α = π/6isalsotheoptimal
commutation angle from the point of minimizing the torque ripple. Moreover,
the equation 4.48 supports the notion introduced earlier in section 4.2.3 that
torque is linearly proportional to the drive current. Therefore, the EM torque
of the motor can be controlled by changing the drive current.
4.4.4.3 Optimal Commutation Angle for CV-BLDC Mode
In the CV-BLDC drive mode, the voltage difference between motor terminals
and ground can be expressed as,
U dc , α≤θ < α+2π/3
0 , α+π≤θ < α+5π/3
e a (θ)+V N , th rs
u A (θ)=
(4.100)
µ
U dc , α−2π/3 ≤θ < α
0 , α+π/3 ≤θ < α+π
e b (θ)+V N , th rs
θ−
3
u B (θ)=u A
=
(4.101)
µ
U dc , α−4π≤θ < α−2π/3
0 , α−π/3 ≤θ < α+ pi/3
e c (θ)+V N , th rs
θ−
3
u C (θ)=u A
=
(4.102)
The currents in the phase winding can be expressed as
i A = u A (θ) −e A (θ) −V N ]/R a
i B = u B (θ) −e B (θ)−V N ]/R a
i C = u C (θ)−e C (θ) −V N ]/R a
(4.103)
It is obvious from these equations that the commutation angle α certainly
affects the drive current, and therefore, affects the EM torque generated in the
motor, which is illustrated by an example in Figure 4.96 showing the torque
produced for two different commutation angle, α=20 and α=50 .
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