Page 501 - Mechanical Engineers' Handbook (Volume 2)

492 Closed-Loop Control System Analysis

Table 6 Optimum Coefficients of T(s) Based on the ITAE
Criterion for a Ramp Input
2 2
n
s 3.2 s n
3 2 2 3
n
n
s 1.75 s 3.25 s n
4 3 2 2 3 4
s 2.41 s 4.93 s 5.14 s n
n
n
n
5 4 2 3 3 2 4 5
s 2.19 s 6.50 s 6.30 s 5.24 s n
n
n
n
n
8.1 Analog Computation 8
An analog computer is a machine in which several physical components can be selected and
interconnected in such a way that the equations describing the operation of the computer are
analogous to that of the actual physical system to be studied. It is a continuous-time device
operating in a real-time parallel mode, making it particularly suitable for the solution of
differential equations and hence for the simulation of dynamic systems. The most commonly
used analog computer is the electronic analog computer in which voltages at various places
within the computer are proportional to the variable terms in the actual system. The ease of
use and the direct interactive control over the running of such a computer allow full scope
for engineering intuition and make it a valuable tool for the analysis of dynamic systems
and the synthesis of any associated controllers. A facility frequently useful is the ability to
slow down or speed up the problem solution. The accuracy of solution, since it is dependent
on analog devices, is generally of the order of a few percent, but for the purposes of system
analysis and design, higher accuracy is seldom necessary; also, this accuracy often matches
the quality of the available input data.
The basic building block of the analog computer is the high-gain dc amplifier, repre-
sented schematically by Fig. 36. When the input voltage is e (t), the output voltage is given
i
by
e (t) Ae (t) (103)
o i
where A, the amplifier voltage gain, is a large constant value.
If the voltage to the input of an amplifier, commonly referred to as the summing junc-
tion, exceeds a few microvolts, then the amplifier becomes saturated or overloaded because
the power supply cannot force the output voltage high enough to give the correct gain.
Therefore, if an amplifier is to be operated correctly, its summing junction must be very
close to ground potential and is usually treated as such in programming.
When this is used in conjunction with a resistance network as shown in Fig. 37, the
resulting circuit can be used to add a number of voltages.
If
R R R
1 2 3
then
V (V V V ) (104)
0 1 2 3
If R , R , R are arbitrary, then
1 2 3
V R ƒ R ƒ V
R
ƒ
1
0
R 1 V R 2 2 R 3 V 3 (105)
If there is only one voltage input, then