Page 707 - Mechatronics with Experiments

ELECTRIC ACTUATORS: MOTOR AND DRIVE TECHNOLOGY 693
c μ l c
A c R =
c c μ A .
lc c
l g
+ R g = μ 0 . g A
l m
N i . R m = . .
lg
N i . r μ μ 0 A m
–
Ag

N S r ϕ = B A .
l m r m
A
m
(a) (b)
FIGURE 8.68: (a) An electromagnetic circuit example involving a permanent magnet, a coil
wound over a core which has an air gap. (b) Magnetic circuit model.

2
rotary inertia (J kg m ) and current to torque gain (K Nm∕A). Let the current amplifier be mod-
m
t
eled by a static gain of voltage command to current gain of K A∕V. The PD controller has gains
a
K and K which define the relationship between the position and velocity error and control signal
d
p
(voltage command). Let the position feedback sensor also be represented by its gain and neglect
2
any dynamic delays, K Counts∕Rad. a) Let J = 10 −4 kg m , K = 0.10 N m∕A, K = 2.0A∕V,
s
a
m
t
K = 2000∕(2 ) Counts∕Rad, K = 0.02 V∕Counts, K = 10 −4 V∕(Counts∕s). Find the loop trans-
p
d
f
fer function (the transfer function from the error signal to the output of the sensor (sensor signal),
then determine the frequency at which the magnitude of the loop transfer function is equal to unity.
At that frequency, determine the phase angle of the loop transfer function. b) Similarly, determine
the frequency at which the phase angle of the loop transfer function is equal to 180 degrees (if such
a finite frequency exists) and find the magnitude of the loop transfer function at that frequency. In a
real DC motor control system, such a frequency does exist even if the above analysis may indicate
that such a frequency is not finite. Discuss why in real hardware such a frequency is finite (Hint:
neglected filtering effects and time delays in the above analysis compared to the real system. A pure
time delay can be modeled in the frequency domain as e −jwt d ,where t is the time delay. It adds phase
d
lag to the loop transfer function).
9. Consider the electromagnetic circuit shown in Figure 8.68 where there are two magnetic field
sources: the permanent magnet and the coil. The coil has N turns and current is i. Let the permeability
constant of the core be . Let the air gap, permanent magnet, core cross-section areas be the same,
c
A = A = A , for simplicity.
c
m
g
(a) Determine the inductance of the circuit and,
(b) the P , magnitude of the slope of load line, assuming =∞.
c
c
(c) Determine the operating point of the permanent magnet under a given non-zero current condition
at the coil. Neglect the reluctance of the core relative to the reluctance of the magnet and air
gap and let the magnet permeability be the same as that of air ( = 1.0). Also, let A = A ,the
m
g
r
cross-section areas of magnet and air gap be the same. Let the coil current be i and the number
of turns of the coil be N.