0195136047.pdf

792 BASIC CONTROL SYSTEMS

E

Feedback

potentiometer

Input

potentiometer

Servo

amplifier

Gain factor Ka

Load

J, F, TL

c

Output

Motor-developed displacement

torque = e KaKm

Direct connection

between servomotor

and feedback

potentiometer

as well as load

Servomotor

(torque constant

= Km)

Actuating signal

e = Kp(r − c)

Er Eb

+

Figure 16.2.7Second-order servomechanism.

Servomotor

Input potentiometer

Feedback potentiometer

−Eb Amplifier

−TL

Js^2 + Fs

R(s) K Er E Ea 1 C(s)

p Ka

Td

Km

Kp

Figure 16.2.8Block diagram of the servomechanism of Figure 16.2.7.

H = 1

−

Js^2 + Fs

REK^1 C

p

Td

KaKm

Figure 16.2.9Simplified block

diagram of the servomechanism

of Figure 16.2.7 (with zero load

torque).

or

M(s)=

C(s)

R(s)

=

K/J

s^2 +(F /J )s+K/J

(16.2.25)

or

C(s)=R(s)

K/J

s^2 +(F /J )s+K/J

=R(s)

ω^2 n

s^2 + 2 ξωns+ωn^2

(16.2.26)

whereωn≡

√

K/Jis known as thesystem natural frequency, and

ξ≡

F

2

√

KJ

=

total damping

critical damping

is known as thedamping ratio.ξandωnare the two figures of merit that describe the dynamic

behavior of any linear second-order system. The damping ratioξprovides information about

the maximum overshoot (see Figure 16.2.6) in the system when it is excited by a step-forcing