082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

21.11 Transfer functions 665
Fuel variation
(mass of fuel
per unit length
of conveyor,
kgm –1 )
d
q(t) q(t – d)
Q(s)
e –sd
Q d (s)
Time (s)
Figure21.14
Thetimedelay,d,isintroducedbytheconveyorbelt.
Figure21.15
Block diagram forthe
conveyor belt.
u(t −d)q(t −d) = quantity of fuelentering the furnace,
d = time delay introduced by the conveyor,
l = length of conveyor (m),
v = speed of conveyor (ms −1 ).
The input to the conveyor is u(t)q(t). The output from the conveyor is
u(t −d)q(t −d). If the conveyor is moving at a constant speed then v = l d and
sod= l . The transfer function that models the conveyor belt can be obtained by
v
using the second shift theorem:
Q d
(s) = L{u(t −d)q(t −d)} = e −sd L{q(t)} = e −sd Q(s)
The transfer function forthe conveyor isshown inFigure 21.15.
Transport lags can cause difficulty when trying to control a system because of
the delay between taking a control action and its effect being felt. In this example,
increasingthequantityoffuelontheconveyordoesnotleadtoanimmediateincrease
in fuel entering the furnace. The difficulty of controlling the furnace temperature
increases as the transport lag introduced by the conveyor increases.
Engineeringapplication21.7
Positioncontrolsystem
There are many examples of position control systems in engineering, for example
controlofthepositionofaplotterpen,andcontrolofthepositionofaradiotelescope.
Thecommon termforthesesystems is servo-systems.
Consider the block diagram of Figure 21.16 which represents a simple servosystem.
The actual position of the motor is denoted by a
(s) in the s domain and
θ a
(t)inthetimedomain.Thedesiredpositionisdenotedby d
(s)and θ d
(t),respectively.
The system is a closed loop with negative feedback. The difference between
the desired and the actual position generates an error signal which is fed to a controllerwithgainK.Theoutputsignalfromthecontrollerisfedtoaservo-motorand
its associated drive circuitry. The aim of the control system is to maintain the actual
position of the motor at a value corresponding to the desired position. In practice, if ➔