Page 651 - Mechatronics with Experiments

ELECTRIC ACTUATORS: MOTOR AND DRIVE TECHNOLOGY 637
hold for magnetic field strength, flux density, and flux itself,
H ⋅ x = n coil ⋅ i (8.130)
g
n ⋅ i
B = ⋅ H = ⋅ coil (8.131)
g
0
g
0
x
n ⋅ i
Φ = B ⋅ A = ⋅ coil ⋅ A g (8.132)
0
b
g
g
x
The flux linkage and the inductance is defined as,
(x, i) =Φ ⋅ n = L(x) ⋅ i(t) (8.133)
b coil
n 2 ⋅ i(t)
= ⋅ coil ⋅ A g (8.134)
0
x
Then, the inductance as a function of plunger displacement is
n 2
L(x) = ⋅ coil ⋅ A (8.135)
0 g
x
The coil has n coil turns, and the voltage is controlled across the terminals of the
coil, V(t). The plunger moves inside in the direction of x. The electromechanical dynamic
model of the solenoid includes three equations: (i) the electromechanical relationship which
describes the voltage, current in coil, and motion of the plunger,
d
V(t) = R ⋅ i(t) + ( (x, i)) (8.136)
dt
where (x, i) is the flux linkage. For an inductor type coil circuit (x, i)is
(x, i) = L(x) ⋅ i(t) (8.137)
Hence, the voltage–current–motion relationship can be expressed as,
dL(x) dx di(t)
V(t) = R ⋅ i(t) + ⋅ ⋅ i(t) + L(x) ⋅ (8.138)
dx dt dt
( )
di(t) dL(x)
V(t) = R ⋅ i(t) + L(x) ⋅ + ⋅ i(t)) ⋅ ̇ x(t) (8.139)
dt dx
The force is calculated from the so-called co-energy equation,
1 1 2
W ( , i) = (x, i) ⋅ i = L(x) ⋅ i (8.140)
co
2 2
W (x, i)
co
F(x, i) = (8.141)
x
1 L(x) 2
= ⋅ i (8.142)
2 x
n 2
1 coil 2
F(x, i) =− ⋅ ⋅ A ⋅ i (8.143)
g
0
2 x 2
Notice that the force direction does not depend on the current and it is proportional to the
square of the current, and inversely proportional to the square of the air gap. The shape of
the force as a function of displacement can be shaped with design of the plunger and stopper
cross-sections (Figure 8.18). Finally, the force–inertia relationship defines the motion of
the plunger and any load it may be driving,
F(t) = m ⋅ ̈ x(t) + k spring ⋅ (x(t) − x ) + F load (t) (8.144)
0
t
where m is the mass of the plunger plus the load, F load is the load force, x is the preload
0
t
displacement of the spring.