Page 696 - Mechatronics with Experiments

682 MECHATRONICS
The electromagnetic force generation between the two components follows the same
physical principles as in the case of their rotary counterparts. Figure 8.57 shows the basic
principle of a linear motor by un-rolling the cylindrical shape to a flat shape. The number
of actively controlled stator phases the same. The commutation of the current in each phase
is based on the cyclic linear distance (pitch) of the permanent magnet dimensions instead
of the rotary angle. In a two pole rotary brushless DC motor, the commutation cycle is 360
degrees, whereas the commutation cycle in a linear brushless motor is the distance between
two consecutive pole pairs (i.e., the distance between two north or two south pole magnets,
total length of a north and a south pole magnet). In general, the same amplifiers used in the
rotary version are used for the linear version of the motor. The feedback sensor (if used for
commutation) is a linear displacement sensor instead of a rotary displacement sensor, that
is Hall effect sensors are used to detect the relative position of stator with respect to the
rotor within a cyclic distance of pitch. Similarly, linear encoders may be used in place of
rotary encoders for position sensing.
Linear brushless DC motors have three basic types: iron core, ironless (air core), and
slotless. In a brushless DC tubular linear motor, the permanent magnets are rolled into a
cylindrical shape around the axis perpendicular to the rotary motor axis. The stator has three
phases and is wound around the rotor. The controller and amplification stages of a linear
motor are identical to those used in the rotary version. The commutation in the amplifier is
based on the cyclic pitch distance of the magnet pairs in the rotor, instead of the angular
position in the case of rotary motors.
The dynamic model of a linear DC motor is identical to that of rotary DC motors.
The electrical dynamics of the motor is
di(t)
V (t) = R ⋅ i(t) + L ⋅ + K ⋅ ̇ x(t) (8.274)
e
t
dt
where V (t) is the terminal voltage, R is the winding resistance, L is the self-inductance, k e
t
is the back EMF gain of the motor. The net electrical power converted to mechanical power
is
P (t) = V (t) ⋅ i(t) (8.275)
e bemf
= K ⋅ ̇ x(t) ⋅ i(t) (8.276)
E
= K ⋅ ̇ x(t) ⋅ i(t) (8.277)
T
= F(t) ⋅ ̇ x(t) (8.278)
= P (t) (8.279)
m
assuming 100% efficiency in converting the electrical power to mechanical power,
P (t) = F(t) ⋅ ̇ x(t) (8.280)
m
= P (t) (8.281)
e
= V (t) ⋅ i(t) (8.282)
bemf
where the force–current relationship is
F(t) = K ⋅ i(t) (8.283)
T
which also shows the force/torque gain is equal to the back EMF gain. The gain K is
t
a function of the magnetic flux (flux density times the cross-sectional area linking the
magnetic field to the conductors) and the number of turns of the coil. In other words, it
is a function of the magnetic flux at the operating point of the permanent magnets and its
size, plus the number of coil turns that links the flux. For a practical motor, the solution
of flux is best obtained by finite element based software tools. Therefore, the force gain is