Direct Torque Control with Space Vector Modulation (DTC-SVM) of ...

Modeling and control modes of PM synchronous motor drives
The instantaneous electromagnetic torque developed by an electric motor can be defined
as:
P
e
M
e
= (2.49)
Ω
m
where, P
e
is the electromagnetic power and
Ωm
is the mechanical angular rotor speed.
Finally, taking into account equation (2.49) the expression for electromagnetic torque
can be obtained as:
and in dqframe: ,
3
∗
Me = pbIm( Ψ
sABC
IsABC
) , (2.50)
2
3
M
e
= pb ( Ψsd Isq −Ψ
sqIsd
)
(2.51)
2
Substituting Ψ , Ψ from (2.33a-b), the torque expression of equations (2.47)
becomes:
sd
sq
3
M
e
= pb ( ΨPM Isq −( Lq − Ld ) Isd Isq
)
(2.52)
2
It can be seen from (2.52), that developed torque consist of two parts, one produced by
the permanent magnet flux called synchronous torque ( M
reluctance torque ( M
er
es
) and the second called
), which is produced by the difference of the inductance in rotor
d- and q-axes. Expressions for those two torque components are:
3
M
es
= pbΨ PM
Isq
(2.53a)
2
3
M
er
=− pb ( Lq − Ld ) Isd Isq
(2.53b)
2
It should be mentioned that for SPMSM ( L d
= L ) the reluctance torque does not exist
q
due to the same inductance paths in rotor d- and q-axes.
The torque expression (2.52) can also be written in polar form using the current vector
amplitude
vector (Fig. 2.5.).
I
s
and the torque angle δ I
, i.e. angle between rotor d-axis and current
19