Case studies for CT systems - Continuous and Discrete Time Signals and Systems - page 379

of the armature. The emf voltage V emf ( t ) is approximated by the product of the
feedback factor k f and the angular velocity ω ( t ).

Motor circuit The torque T m , induced by the applied voltage across the arma-
ture, is given by

T m = k m i a ( t ) , (8.30)

where k m is referred to as the motor or armature constant and i a ( t ) is the armature
current. The armature constant k m depends on the physical properties of the dc
motor such as the strength of the magnetic field and the density of the armature
coil.

Load The load component of the dc motor is obtained by applying Newton's
third law of motion, which states that the sum of the applied and reactive forces
is zero. The applied forces are the torques around the motor shaft. The reactive
force causes acceleration of the armature and equals the product of the inertial
load J and the derivative of the angular rate ω ( t ). In other words,

J d ω
d t

T p = , (8.31)

p

where J denotes the inertia of the rotor. There are three different torques, i.e.
p = 3, observed at the shaft: (i) motor torque T m represented by Eq. (8.30); (ii)
frictional torque T f given by r ω ( t ), r being the frictional constant; and (iii) load
disturbance torque T d . In other words, Eq. (8.31) can be expressed as follows:
J d ω
d t

= T m − r ω ( t ) − T d . (8.32)

Since the angular velocity ω ( t ) is related to the shaft position θ ( t ) by the fol-
lowing expression:

ω ( t ) = d θ

d t ,

(8.33)

Eq. (8.32) can be expressed as follows:
J d 2 θ
d t 2

+ r d θ
d t

= T m − T d = T L , (8.34)

where T L denotes the difference between the motor torque T m and the load
disturbance torque T d .

8.3.2 Transfer function

The dc motor shown in Fig. 8.8 is modeled as a linear time-invariant (LTI)
system with the armature voltage v a ( t ) considered as the input signal and the
shaft position θ ( t ) as the output signal. We now derive the transfer function of
the linearized model.