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Potentiometer,
servomotor, and tachogenerators |
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Problem 11.1 Transfer function of a d.c servo-motor
(a) Find an expression relating the applied voltage v (t)
with the angular position q(t)
for the
armature controlled dc servomotor shown in Fig. 1.
J, D
Fig.1
R=
armature resistance
J =
moment of inertia of the moving parts
D=
damping coefficient due to friction
ω
(t) =d q(t)/dt=angular velocity
vb
(t) = motor back emf= Kb ω (t)
Kb
= motor back emf constant
Tm
(t) = motor torque =Km i (t)
i (t)
= armature current
Km
= motor torque constant.
(b) Draw an s-domain block diagram showing the explicit
relationships among the variables V(s), Vb(s), I(s), Tm(s),
ω(s) and q(s).
Solution
(a)
i (t) = [v (t)-vb
(t)]/R
TL= Jdw/dt +Dw
Equate Tm=TL and obtain
Km (v-Kbw)/R = Jdw/dt + Dw
Rearrange,
dw/dt + (D + KmKb/R) w/J = Kmv/JR
Or,
dw/dt + aw =Kv (1)
where
a = (D + KmKb/R)/J
and K = Km/JR
Eqn.1 may be written as
d2q/dt2
+ a. dq/dt
=K.v
(b)
Take
q(s)/V(s) = K/s(s+a)
1/s 1/ (Js+D) Km 1/R
Kb
-
Vb(s)
Problem 11.2 Potentiometer, servomotor, and tachogenerators
(a)State at
least three disadvantages of the potentiometer transducer.
(b) State at least two desirable requirements of a
servomotor.
(c) Explain with neat sketches the working of the
following tacho-generators:
I.
D.C tacho-generator
II.
Single-phase a.c tacho-generator with bridge rectifier.
III.
Three-phase a.c tacho-generator with bridge rectifier.
(d)
Show that the transfer function of the network of Fig.2 is of the form:
E0(s)/Ei(s) = st /
(1+ st)
ei R eo
Fig.2
Define t in terms of the network
parameters.
Solution
(a)
(i) The potentiometer wires are extremely delicate and the wiper should be handled with care
(ii) Output voltage is discontinuous, deteriorating servo accuracy.
(iii)Relatively large force is required to move the contact arm and a relatively large displacement is necessary for the wiper
(iv)Potentiometer may be use when the displacements are of a limited nature. Limit protection is essential and expensive in many cases.
(v) Sensitivity is limited because heat dissipation limits use of higher applied voltage. A relatively high gain amplifier is required.
(vi) Transducer is unsuitable for systems where extremely small displacement is to be sensed for control purposes
(b)(i) rotor should have a low moment of inertia
(ii) speed-torque curve should have a negative slope, which should be the same over the entire range of voltage regulation
(i) Motor should withstand frequent starting operations
(c) Descriptive
(d) t= RC
Problem 11.3 D.C
motor transfer function
(a) List the major advantages and
disadvantages of open-loop control systems.
(b) A d.c motor has a fixed field
excitation. A voltage V is applied through an amplifier of gain K, as shown in
Fig.3.
K
Fig.3
R=armature resistance
L=armature inductance
E= back-e.m.f
Ke= back-e.m.f constant
KT= torque constant
T = torque
J= rotor moment of inertia
F=coefficient of friction
q= shaft angle
Derive the transfer function, q(s)/V(s).
Solution
(a) Advantages
1. Simple construction & ease of maintenance
2. Less expensive than a corresponding closed
–loop system
3. No stability problem
4. Convenient when output is hard to measure
1. Disturbances &
changes in calibration cause errors and the output may be different from what is
desired.
2. Re-calibration
necessary from time to time.
(b) The transformed differential equations are:
(i)
Kirchhof’s
law
V(s) =IR +sLI + Kb sq------(A)
(ii)
KTI= Js2q+fsq------- (B)
Eliminate I from (A) and (B) and re-arrange
q(s)/V(s) = KKT/ís[ JLs2
+s (JR+fL) +fR+Kb KT]ý
(i)
The servo motor
differs from the standard motor principally in that
(a)
it has entirely
different construction
(b)
it has high inertia
and hence high torque
(c)
it has low inertia
and low torque
(d)
it has low inertia
and high starting torque.
(e)
Ans: (d)
(ii)
The most common use
of synchro is
(a)
as error detector
(b)
for transmission of angular data
(c)
for transmission of
arithmetic data
(d)
for all the above purposes.
Ans: (b)