Page 559 - Mechanical Engineers' Handbook (Volume 2)

550 Servoactuators for Closed-Loop Control

Taking Laplace transforms of Eqs. (1)–(3) gives, after algebraic manipulation,
(s) G (s)V (s) G (s)[T (s) T (s)] (4)
m
2
a
ƒm
L
1
where the transfer functions G and G are given by
1
2
K
G (s) T (5)
1
RB ( s 1)( s 1) KK E
T
e
am
m
R ( s 1)
G (s) a e (6)
2
RB ( s 1)( s 1) KK
am e m T E
and the parameters are defined as follows:
B viscous damping in motor (N m s/rad)
m
e back electromotive force (emf) (V)
b
i current through armature (A)
J polar moment of inertia of armature (N m s /rad)
2
m
K z P/60 motor voltage constant or back emf constant (V s/rad)
E
K z P/2 motor torque constant (N m/A)
T
L armature inductance (H)
a
P number of poles
R armature resistance ( )
a
s Laplace operator
t time (s)
T ƒm Coulomb friction torque in motor (N m)
T external load torque
L
v voltage applied to armature (V)
a
V (s) Laplace transform of armature voltage v (t)
a
a
z number of conductors per parallel path in armature
angular position of motor shaft (rad)
m
magnetic flux per pole (Wb)
L /R electrical time constant (s)
a
a
e
J /B mechanical time constant (s)†
m
m
m
angular velocity of motor (rad/s)
m
(s) laplace transform of motor angular velocity
m
Equation (4) can be simplified if the armature inductance is small (making the electrical time
constant negligible) and the Coulomb friction and load torque are assumed zero. The
e
result is
(s) K m (7)
m
V (s) s 1
a
where
K
K T (motor constant) (8)
m
RB KK E
am
T
and
9
*Some servomotor manufacturers define m differently. For example, Electro-Craft defines the mechan-
ical time constant as m (R a J m )/(K T K E ).