On the other hand, I-PMSMs show a salient magnetic structure; thus, their inductances can be written as:
L
qs
= L
Is
+
3
2
L
ms
+ L
2s
L
ds
= L
is
+
3
2
L
ms
− L
2s
(6)
SM-PMSM field-oriented control (FOC)
The equations below describe the electromagnetic torque of an SM-PMSM:
T
e
=
3
2
p λ
ds
i
qs
− λ
qs
i
ds
=
3
2
p L
s
i
ds
i
qs
− L
s
i
qs
i
ds
+ Φ
m
i
qs
(7)
T
e
=
3
2
p Φ
m
i
qs
(8)
The last equation makes it clear that the quadrature current component
i
qs
has linear control on the torque
generation, whereas the current component
i
ds
has no effect on it (as mentioned above, these equations are valid
for SM-PMSMs).
Therefore, if
I
s
is the motor rated current, then its maximum torque is produced for
i
qs
=
I
s
and
i
ds
= 0
(in fact,
I
s
= i
qs
2
+ i
ds
2
). In any case, it is clear that, when using an SM-PMSM, the torque/current ratio is optimized by
letting
i
ds
= 0
. This choice corresponds to the MTPA (maximum-torque-per-ampere) control for isotropic motors.
On the other hand, the magnetic flux can be weakened by acting on the direct axis current
i
ds
; this extends the
achievable speed range, but at the cost of a decrease in maximum quadrature current
i
qs
, and hence in the
electromagnetic torque supplied to the load.
In conclusion, by regulating the motor currents through their components
i
qs
and
i
ds
, FOC manages to regulate
the PMSM torque and flux. Current regulation is achieved by means of what is usually called a “synchronous
frame CR-PWM”.
3.1.3 PID regulator theoretical background
The regulators implemented for Torque, Flux and Speed are actually Proportional Integral Derivative (PID)
regulators. PID regulator theory and tuning methods are subjects which have been extensively discussed in
technical literature. This section provides a basic reminder of the theory.
PID regulators are useful to maintain a level of torque, flux or speed according to a desired target. Indeed, both
the torque and the flux are a function of the rotor position. FOC needs to regulate torque and flux to maximize
system efficiency. In addition, the torque is also a function of the rotor speed. Hence, performing speed regulation
results into regulating the torque.
The following is the PID general equation. it is used in the general PID regulator:
r t
k
= K
p
× ϵ t
k
+ K
i
× ∑
j = 0
k
ϵ
t
j
+K
d
× ϵ t
k
− ϵ t
k − 1
(9)
where:
•
ϵ t
k
is the error of the system observed at time
t = t
k
, while ϵ t
k − 1
is the error of the system at time
t = t
k
− T
sampling
•
K
p
is the proportional coefficient.
•
K
i
is the integral coefficient.
•
K
d
is the differential coefficient.
•
r t
k
is the reference to apply as output of the PID regulator
In a motor control application, the derivative term can be disabled. This is indeed the case in the STM32 Motor
Control SDK although both PID regulator implantations are provided.
3.1.4 Regulator sampling time setting
The sampling period T
sampling
needs to be modified to adjust the regulation bandwidth. As an accumulative term
(the integral term) is used in the algorithm, increasing the loop time decreases its effects (accumulation is slower
and the integral action on the output is delayed). Inversely, decreasing the loop time increases its effects
(accumulation is faster and the integral action on the output is increased). This is why this parameter has to be
adjusted prior to setting up any coefficient of the PID regulator.
UM2392
Introduction to PMSM FOC drive
UM2392 - Rev 1
page 11/60