Page 103 - Servo Motors and Industrial Control Theory -

96 5 Stepping Servo Motors

The second term in Eq. (5.4) is negative because as θ increases over one step of
stepping motor, the torque reduces. When θ moves one step, the torque becomes
zero. The torque constant ( K ) and the step angle are given by manufacturers. There-
t
fore, K can easily be evaluated.
θ

5.4 Dynamic Response Characteristic over One Step
Movement

It is important first to study the dynamic response of stepping motors for single step
movement. To do this the linearized Eq. (5.4) must be used. In addition several as-
sumptions have to be made. For the simplest model, the inductance of the windings
is ignored. The voltage equation simply becomes
V : = RI (5.7)

Eliminating the variable I from Eqs. (5.4) and (5.7) gives
K
−
T: = t ⋅ V K ⋅θ (5.8)

m θ
R
The equation of motion for the rotor assuming the total inertia of J and a viscous
mechanical damping of C may be written as
T : = Js θ +Cs +TIθ (5.9)
2
m
where T is the external torque applied to the motor. Eliminating the variable T
l
m
from the two Eqs (5.8) and (5.9) and with some algebraic manipulations, the trans-
fer function becomes

K
−
t ⋅ VT
RK 1
: θ= θ (5.10)
J s ⋅ 2 + C s +θ
K θ K θ

It can be seen that the governing transfer function in simplest form is a second order
with two input variables of voltage and external torque. The coefficients of the char-
acteristic equation gives the natural frequency and the damping ratio as

1 : = J ω : = K θ
ω n 2 K θ n J (5.11)