Modern Control Engineering

96 Chapter 3 / Mathematical Modeling of Mechanical Systems and Electrical Systems

Obtain the transfer function between the motor shaft angular displacement uand the error

voltageev.Obtain also a block diagram for this system and a simplified block diagram when La

is negligible.

Solution.The speed of an armature-controlled dc servomotor is controlled by the armature volt-

ageea.(The armature voltage is the output of the amplifier.) The differential equation

for the armature circuit is

or

(3–46)

The equation for torque equilibrium is

(3–47)

whereJ 0 is the inertia of the combination of the motor, load, and gear train referred to the motor

shaft and b 0 is the viscous-friction coefficient of the combination of the motor, load, and gear train

referred to the motor shaft.

By eliminating iafrom Equations (3–46) and (3–47), we obtain

(3–48)

We assume that the gear ratio of the gear train is such that the output shaft rotates ntimes for each

revolution of the motor shaft. Thus,

(3–49)

The relationship among Ev(s), R(s),andC(s)is

(3–50)

The block diagram of this system can be constructed from Equations (3–48), (3–49), and (3–50),

as shown in Figure 3–29(b). The transfer function in the feedforward path of this system is

When Lais small, it can be neglected, and the transfer function G(s)in the feedforward path

becomes

(3–51)

The term indicates that the back emf of the motor effectively increases the

viscous friction of the system. The inertia J 0 and viscous friction coefficient b 0 +AK 2 K 3 RaBare

Cb 0 +AK 2 K 3 RaBDs

=

K 0 K 1 K 2 nRa

J 0 s^2 + ab 0 +

K 2 K 3

Ra

bs

G(s)=

K 0 K 1 K 2 n

sCRaAJ 0 s+b 0 B+K 2 K 3 D

G(s)=

C(s)

Q (s)

Q (s)

Ev(s)

Ev(s)

E(s)

=

K 0 K 1 K 2 n

sCALa s+RaBAJ 0 s+b 0 B+K 2 K 3 D

Ev(s)=K 0 CR(s)-C(s)D=K 0 E(s)

C(s)=nQ (s)

Q (s)

Ev(s)

=

K 1 K 2

sALa s+RaBAJ 0 s+b 0 B+K 2 K 3 s

J 0

d^2 u

dt^2

+b 0

du

dt

=T=K 2 ia

La

dia

dt

+Ra ia+K 3

du

dt

=K 1 ev

La

dia

dt

+Ra ia+eb=ea

ea=K 1 ev

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