Datasheet

For the magnetic field equations, we got:
)sin(
)cos(




mm
mm
LiLi
LiLi


After performing the derivation:
mm
mm
dt
di
L
dt
di
L
dt
d
dt
di
L
dt
di
L
dt
d










)cos(
)sin(
Finally, we obtain for the voltages in (,) system:
mS
mS
dt
di
LiRv
dt
di
LiRv










A second reference frame is used to represent the equations as the frame is turning at the rotor speed. So the “d” axis is
chosen in the direction of the magnetic vector 
m
, and with the “q” axis orthogonal to the “d” axis. The new reference
system is (d, q).
The reference frame transformations from the (,) system to the (d, q) system depends on the instantaneous position
angle .
So we obtain two inter-dependant equations in the (d, q) system:
md
q
qSq
q
d
dSd
Li
dt
di
LiRv
Li
dt
di
LiRv




These two equations represent the mathematical motor model.
Vd and Vq Equation Diagram
A control algorithm which wants to produce determined currents in the (d, q) system must impose voltages given from
the formulas above. This is ensured by closed loop PI control on both axis “d” and “q” (Proportional Integral).
Since there is a mutual influence between the two axes, decoupling terms can be used.
1/(R+sL)
1/(R+sL)
(3/2)p 
1/(B+sJ)
 load
p 
pL
V d
V q
+
-
+
- -
+
 mec
I d
I q 
+
L e
 e