Page 178 - Electric Drives and Electromechanical Systems

172 Electric Drives and Electromechanical Systems

The factor of two in this equation is the result of the current simultaneously flowing
through two motor phases. The electromagnetic torque can be determined from Eqs
(6.4) and (6.5),
T e ¼ 4N P B g lrI (6.6)
Rewriting the voltage and torque equations with E ¼ 2e p , to represent the emf of any
two phases in series, Eq. (6.4) and Eq. (6.6) can be rewritten as,

0 (6.7a)
E ¼ kju m def K u m
v
0
T e ¼ kjI def K I (6.7b)
T
where the armature constant, k ¼ 4N P , and the flux, j ¼ B g pirl, are determined by the
construction of the motor and its magnetic properties. The form of these two equations
is very similar to the corresponding equations for brushed d.c. motors (see Eq. 5.1); this
explains, to a large extent, why these motors are called brushless d.c. motors within the
drives industry, when they are more correctly described as permanent-magnet syn-
chronous motors with a trapezoidal flux distribution. In practice, the equations above
will only hold good if the switching between the phases is instantaneous, and if the flux
density is uniform with no fringing; while this does not hold true for real motors, these
equations can be safely used during the normal selection procedure for a motor and its
associated controller.

6.1.1 Torque-speed characteristics

Using the relationships above, the torque-speed characteristics of an ideal BDCM can be
determined; it is assumed that the commutation and back emf voltage waveforms are
perfect, as shown in Fig. 6.5. If the star-connected motor configuration is considered, the
instantaneous voltage equation can be written as,
V s ¼ E þ IR (6.8)
where R is the sum of the individual phase resistances, V s is the motor’s terminal voltage
(neglecting semiconductor and other voltage drops), and E is the sum of the emf’s for two
phase. Using the voltage, torque, and speed equations discussed above, the motor’s
torque-speed characteristics can be determined. The torque-speed relationship is given
by,
T

u m ¼ u 0 1 (6.9)
T 0
where the no-load speed is defined by,
V s
u 0 ¼ (6.10)
kj
The stall torque of the motor is given by,
(6.11)
T 0 ¼ kjI 0
where,
V s
I 0 ¼ (6.12)
R