Page 79 - Servo Motors and Industrial Control Theory -

72 4 Electrical DC Servo Motors

4.5 DC Servo Motors in Closed Loop Position Control

First, a proportional control in which the command signal with the output position
is compared and the difference is multiplied by a gain is considered. The equation
for the control unit is
V : K( = θ −θ o (4.1)
)
i
The voltage equation of the motor ignoring the inductance becomes

V : RI+C s = θ (4.2)
m o
The second term in Eq. (4.2) is the internal velocity feedback. The effect of induc-
tance will be considered for very high performance applications.
As mentioned before the torque produced by the motor is proportional to the
current as
T := K I (4.3)
t
The torque then accelerates the total inertia attached to the motor and assuming that
the transmission mechanism is very stiff, the equation of motion becomes
2
T : Js + Cs T= θ o θ+ 1 (4.4)
o
The external torque is positive because it usually opposes the motion.
For complicated systems, it is often easier to derive the governing transfer func-
tion directly from the governing differential equations and it is not necessary to
draw the block diagram of the system. The readers are encouraged to drive the
overall transfer function with both methods.
Eliminating V, I, and T from Eqs. (4.1– 4.4) and with some complicated algebraic
manipulation, the transfer function of the systems becomes

θ− R ·T
:= i (KK ) t 1 (4.5)
θ
o JR s + R C+ (C K ) t s+1
2
m
KK t KK t R
It can be seen that there are two inputs variables of θ ,T and one output variable. The
l
i
principle of superposition may be used to study the effect of both input variables.
When the external torque is set to zero and at steady state for step input of θ , the
i
Laplace Transform s may be set to zero. This shows that the output is exactly the
same as input meaning that the steady state error is zero. For a ramp input, the er-
ror must be calculated and it can be shown that there will be a following error. The
reader is encouraged to calculate the following error with procedures discussed in
previous chapters.
When the external torque is applied, a steady state is generated and steady state
error is given by