Modern Control Engineering

Example Problems and Solutions 95

A–3–9. Consider the servo system shown in Figure 3–29(a). The motor shown is a servomotor, a dc motor de-

signed specifically to be used in a control system. The operation of this system is as follows: A pair of

potentiometers acts as an error-measuring device. They convert the input and output positions into

proportional electric signals. The command input signal determines the angular position rof the

wiper arm of the input potentiometer. The angular position ris the reference input to the system, and

the electric potential of the arm is proportional to the angular position of the arm. The output shaft

position determines the angular position cof the wiper arm of the output potentiometer. The differ-

ence between the input angular position rand the output angular position cis the error signal e,or

The potential difference is the error voltage, where eris proportional to randecis pro-

portional to c; that is, and where K 0 is a proportionality constant. The error volt-

age that appears at the potentiometer terminals is amplified by the amplifier whose gain constant is K 1.

The output voltage of this amplifier is applied to the armature circuit of the dc motor. A fixed volt-

age is applied to the field winding. If an error exists, the motor develops a torque to rotate the out-

put load in such a way as to reduce the error to zero. For constant field current, the torque

developed by the motor is

whereK 2 is the motor torque constant and iais the armature current.

When the armature is rotating, a voltage proportional to the product of the flux and angular

velocity is induced in the armature. For a constant flux, the induced voltage ebis directly propor-

tional to the angular velocity or

whereebis the back emf,K 3 is the back emf constant of the motor, and uis the angular displace-

ment of the motor shaft.

eb=K 3

du

dt

dudt,

T=K 2 ia

er=K 0 r ec=K 0 c ,

er-ec=ev

e=r-c

(a)

Reference input

Input device

Input potentiometer

Output potentiometer

Feedback signal

er

ec

r c

c

K 1

ia T

Ra La

Error measuring device Amplifier Motor Gear

train

Load

u

ev K 1 ev

(b) (c)

R(s) E(s) Ev(s) K 1 K 2 U(s) C(s)

K (^0) s(Las+Ra) (Jos+bo)+K 2 K 3 s n
R(s) K C(s)
+– +– s(Js+B)
Figure 3–29
(a) Schematic diagram of servo system; (b) block diagram for the system; (c) simplified block diagram.