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control_5.pptx
1. Chapter Five
Introduction to Analogue Computing Techniques
Topics to be covered
Introduction
Components of the analog computer
Simulation of Linear Systems
Magnitude and Time Scaling
2. Cont..
Objective
The objective of learning this topic is realize the differential equation
using (analog computer) operational amplifier.
3. 5.1. Introduction
Before digital computers became so universal, analog computers were
popular for solving problems such as differential equations.
The basic building block of the analog computer is the operational
amplifier.
In addition to this, analog computers consists of resistor, capacitor and
inductor.
The ideal op-amp has the following properties:
1. The voltage between the +ve and the –ve terminals is zero. That is 𝑒+
= 𝑒−
. This property is
called the virtual ground or the virtual short.
2. The input impedance is infinite. This implies that, the currents into the +ve and the –ve input
terminals is zero.
3. The output impedance of an ideal op-amp is very low i.e., 𝑧𝑜𝑢𝑡 = (50 𝑢𝑝 𝑡𝑜 75)Ω.
4. 5.2. Components of The Analog Computer
i. Inverting Amplifier:
𝑖1 = 𝑖2
⟹
0−𝑣𝑖
𝑅1
=
𝑣𝑜−0
𝑅2
⟹
−𝑣𝑖
𝑅1
=
𝑣𝑜
𝑅2
⟹
𝑣𝑜
𝑣𝑖
= −
𝑅2
𝑅1
• If 𝑅1 = 𝑅2, then
𝑣𝑜 = −𝑣𝑖
10. 5.3. Simulation of Linear Systems
The procedures for obtaining an operational amplifier realization of the linear
systems are:
1. Find the mathematical model of the system.
2. Represent the mathematical model in the analog block diagram.
• Some of the analog components symbols;
3. From the analog block diagram, construct the circuit diagram by using op-amps, resistors
and capacitors.
4. Operate the computer and observes the output variables.
11. Cont’d…
Example: Realize the following dynamic equations by using analog components.
i) 𝑦 = −K𝑥 where K > 0
• We will use an inverting amplifier.
ii) 𝑦 = K𝑥
12. Cont’d…
• We will use two inverting amplifiers in series.
iii) 𝑦 = 𝑥1 + 𝑥2 + 𝑥3
13. Cont’d…
• We will use a summer and an inverting amplifier in series.
iv) 𝑥 = 𝑥
14. Cont’d…
• It uses an integrator in series with an inverter.
where RC = 1
v) 𝑣𝑜 = −𝑣1 − 2𝑣2 + 3𝑣3 + 4𝑣4
From this
𝑣𝑜 = −[𝑣1 + 2𝑣2 − 3𝑣3 − 4𝑣4]
20. Exercise:
1. Find the op-amp realization and analog block diagram for the following.
𝑥 =
1 −1 1
1 −1 1
2 2 1
𝑥 +
0
0
1
𝑢
𝑦 = 1 1 2 𝑥 − 4𝑢
2. Develop a state variable and output equation for the operational amplifier circuit
shown in below
21. 5.4. Magnitude and Time Scaling
Magnitude Scaling:-
It is often the case that the solution to a system differential equation in real time
varies too fast or too slowly to be captured by CRT or plotter. Examples of such
instance are chemical reaction and evaluation of state where time scaling may be
necessary.
Almost any problem where magnitude variation is either too small or too large
needs magnitude scaling.
Consider the equation of motion
𝑀𝑥 + 𝐷𝑥 + 𝐾𝑥 = 𝑓(𝑡)
𝑥 =
1
𝑀
𝑓 𝑡 −
𝐷
𝑀
𝑥 −
𝐾
𝑀
𝑥 (1)
By using an estimation or some other means, let us put a bounds on the variables as
the following
22. Cont’d…
𝑥 ≤ 𝑥𝑚 (𝑚𝑒𝑡𝑒𝑟)
𝑥 ≤ 𝑥𝑚 (𝑚𝑒𝑡𝑒𝑟/𝑠𝑒𝑐)
𝑥 ≤ 𝑥𝑚 (𝑚𝑒𝑡𝑒𝑟/𝑠𝑒𝑐2)
𝑓 ≤ 𝑓𝑚
• And also, let the maximum voltage on the computer be 𝑉
𝑚 = 10V. By using
this maximum voltage, the following scale factors are introduced.
𝐾𝑥 =
10
𝑥𝑚
𝑣 𝑚, 𝑣 𝑚 means 𝑣𝑜𝑙𝑡 𝑚𝑒𝑡𝑒𝑟
𝐾𝑥 =
10
𝑥𝑚
𝑣 𝑚 𝑝𝑒𝑟 𝑠𝑒𝑐
23. Cont’d…
𝐾𝑥 =
10
𝑥𝑚
𝑣 𝑚 𝑝𝑒𝑟 𝑠𝑒𝑐2
𝐾𝑓 =
10
𝑓𝑚
𝑣 𝑁𝑒𝑤𝑡𝑜𝑛
• Then, Equation (1) can be written as the following;
[𝐾𝑥𝑥]
𝐾𝑥
=
[𝐾𝑓𝑓]
𝑀𝐾𝑓
−
𝐷[𝐾𝑥𝑥]
𝑀𝐾𝑥
−
𝐾[𝐾𝑥𝑥]
𝑀𝐾𝑥
[𝐾𝑥𝑥] =
𝐾𝑥
𝑀𝐾𝑓
[𝐾𝑓𝑓] −
𝐷𝐾𝑥
𝑀𝐾𝑥
[𝐾𝑥𝑥] −
𝐾𝐾𝑥
𝑀𝐾𝑥
[𝐾𝑥𝑥]
• The main point is that the terms in the square bracket that are called machine
variables, are limited with in ±10𝑉.
25. Cont’d…
Time Scaling:
Consider the following equation of the motion
𝑥(𝑡) + 𝑎𝑥 𝑡 + 𝑏𝑥 𝑡 = 𝑓(𝑡) (2)
• Time scaling can be introduced by using change of variables.
• Let
𝜏 = 𝛼𝑡
where
0 < 𝛼 < 1 ⇝corresponding for speeding up the system operation time.
1 < 𝛼 < ∞ ⇝corresponding for slow down the system operation time.
Then
𝑑𝜏 = 𝛼𝑑𝑡 ⟹ 𝑑𝑡 = 1
𝛼𝑑𝜏
29. Cont’d…
Example: Consider the following equation of motion
𝑥 =
1
100
𝑓 𝑡 −
1
100
𝑥 −
1
100
𝑥
• If the bounds on each elements (states) and the maximum operational voltage
range of the system are given by below, find the scale factors, the machine
variables and the analog and op-amp realization of the magnitude scaled system
dynamic model.
𝑥 ≤ 𝑥𝑚 , 𝑥𝑚 = 100𝑚
𝑥 ≤ 𝑥𝑚 , 𝑥𝑚 = 10𝑚/𝑠
𝑥 ≤ 𝑥𝑚 , 𝑥𝑚 = 2𝑚/𝑠2
𝑓 ≤ 𝑓𝑚 , 𝑓𝑚 = 100𝑁𝑒𝑤𝑡𝑜𝑛
𝑣 ≤ 𝑣𝑚 , 𝑣𝑚 = 10V
