Page 14 - Servo Motors and Industrial Control Theory

6 1 Feedback Control Theory

Example 3 Speed of DC motors assuming negligible inductance.
In this example, only a brief discussion on modeling of DC motors will be given.
Full details of analysis will be discussed in a different chapter.
The voltage equation can be written as
v = Ri C ω+ m m (1.16)
i

the torque current relation and the equation of motion become
dω
T = J m (1.17)
m
dt

T = K i (1.18)
m t

The parameters K ,C are the torque and voltage constants of the motor respectively.
m
t
Eliminating T ,i from the three equations gives
m
RJ d ω
v = m + C ω (1.19)
i
K t dt m m
Taking Laplace Transform from both sides of Eq. (1.19) and referring to Table 1.1
yields
RJ
() =
Vs sω m () s + C ω m () s (1.20)
m
i
K
t
rearranging gives
1
ω () s = Vs
()
m RJ sC+ i (1.21)
K t m

now Eq. (1.21) can be solved for any input function, for a step input of V, where
V
LV() = , using partial fraction method, we have
s
 1 τ − 1
ω m ()s =  +  V (1.22)
s
 C τ + 1 s 
m
Taking the inverse of Laplace Transform of Eq. (1.22) gives
V
ω ( )t = (1 e− − tτ ) (1.23)
m
C m