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MODULE TITLE: PROGRAMMABLE LOGIC CONTROLLERS

TOPIC TITLE: INTRODUCTORY BASIC THEORY

LESSON 1: BOOLEAN FUNCTIONS AND SYMBOLS

PLC - 1 - 1

Published by Teesside University Open Learning (Engineering)

School of Science & Engineering

Teesside University

Tees Valley, UK

TS1 3BA

+44 (0)1642 342740

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________________________________________________________________________________________

INTRODUCTION ________________________________________________________________________________________

Before we approach the topics of programmable logic controllers (PLCs),

which are essentially programmable computers, it will be useful to familiarise

ourselves with some topics that will later help in our understanding of PLCs.

________________________________________________________________________________________

YOUR AIMS ________________________________________________________________________________________

At the end of this lesson you should be able to:

• understand the meaning of various Boolean operators

• recognise Boolean logic symbols

• understand how basic Boolean operators can be used to form

Boolean functions or expressions

• compose truth tables for a variety of Boolean operators and functions

• use truth tables to identify equivalent Boolean expressions

• use truth tables to simplify Boolean expressions.

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________________________________________________________________________________________

BOOLEAN ALGEBRA ________________________________________________________________________________________

Boolean algebra, or possibly more aptly, Boolean logic or the science of logic,

was invented by the Englishman George Boole. In 1854 he published

‘Investigation of the Laws of Thought’, on which are founded the mathematical

theories of logic and probability. This paper, following work by De Morgan

and others, allied logic to mathematics and destroyed the notion (not without

controversy) that logic was the sole preserve of the realm of philosophy.

Boolean logic, little more than a curiosity for almost a century, was to become

essential to the design and operation of electronic computers. Two important

innovations of Boole were:

(i) the establishment of a mathematical framework for transposing

verbal logical propositions into mathematical symbols

(ii) the extension of numbers and algebraic symbols to represent mental

propositions. This led to further laws not present in classical algebra.

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TRANSPOSITION OF LOGICAL STATEMENTS INTO SYMBOLIC LOGIC

Some statements can be analysed and represented as algebraic variables,

usually with one dependent and the rest independent. The statement:

‘He goes to bed if he has finished studying or is tired’

can be represented by symbols as:

F = A or B where F represents ‘He goes to bed’

(the dependent variable)

A represents ‘he has finished studying’

B represents 'is tired'

(A and B are independent variables).

This statement is an example of an ‘OR’ function, one of the new functions

invented by Boole, and is represented by a ‘+’ (and sometimes a ‘∨’), which should not be confused with the plus or addition sign of normal arithmetic or

algebra. The above statement can be represented as:

F = A + B

We sometimes refer to this ‘OR’ function as the ‘logical sum’.

Another example of a different logical function is:

‘The television will operate if the set is switched on and the aerial is

plugged in’.

F (the television will operate) = A (the set is switched on)

and B (the aerial is plugged in)

This illustrates the AND function.

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In Boolean logic, the AND function is represented as ‘.’ (and sometimes as

‘∧’) so the AND function on the previous page can be written:

F = A.B and is sometimes referred to as the logical product.

(Note: In many situations the dot is omitted for convenience in writing down

expressions, and the function would be written simply as F = AB)

If we consider the statements:

‘The examination was passed’, represented by ‘A’

and

‘The examination was failed’, represented by ‘B’

we can also represent this latter statement as the opposite or complement of A,

that is, B is ‘not A’.

This is normally represented as:

or sometimes as B = ~A

This is called a NOT function.

Combination of these three functions gives rise to other functions, for

example:

OR followed by NOT = NOT OR, called a NOR function

AND followed by NOT = NOT AND, called a NAND function.

B A=

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One other function is commonly used and that is the EXCLUSIVE OR (often

written EXOR or XOR). This can also be combined with a NOT function to

produce an EXCLUSIVE NOR function (often written EXNOR or XNOR).

We will return to these functions later in this lesson.

________________________________________________________________________________________

TRUTH TABLES ________________________________________________________________________________________

The validity of a statement is conveniently examined by means of a table

called a truth table. As the outcome of certain statements can only be either

true or false, all possibilities can be listed, as for example in the statement:

‘He goes to bed if he has finished his studies or is tired’.

F = A + B

The truth table for this OR function is shown in TABLE 1 below:

Note that F is true if either A or B

is true or both A and B are true.

TABLE 1

As there are only two possibilities for any variable, then it is convenient to

assign the state true as ‘1’ and the state false as ‘0’. One of the reasons for this

is that the binary number system can be applied to such a two state or two

number system.

true

false

true

false

true

false

F = A + B

true

false

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Two-state electronic devices called gates have also been developed, which can

be used to simulate Boolean expressions.

Replacing ‘true’ and ‘false’ by ‘1’ and ‘0’ respectively, the above truth table

becomes that shown in TABLE 2:

TABLE 2

The AND function truth table is shown in TABLE 3:

Note F is true only when A

and B are both true.

TABLE 3

F = A . B

F = A + B

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________________________________________________________________________________________

EXAMPLES OF BOOLEAN FUNCTIONS IN REAL LIFE ________________________________________________________________________________________

If we consider the piping arrangement in FIGURE 1 it is clear that the turbine

will only operate if valve A AND valve B are both open, that is:

FIG. 1

Turbine operates F = A . B

Often valves will have bypass valves in case of faulty operation of a valve or

the need to remove a valve for maintenance. FIGURE 2 shows a more usual

arrangement.

FIG. 2

Reservoir Control valve

Outlet valve

Turbine

A B

C D

Reservoir Control valve

Outlet valve

Turbine

A B

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Write down the Boolean expressions for the operation of the turbine with the bypass

arrangements.

....................................................................................................................................................

________________________________________________________________________________________

For the turbine to operate (F) either valve A or C must be open and valve B or

D must be open, that is:

F = (A + C).(B + D)

Another example is the electrical circuit shown in FIGURE 3. The two

vertical lines represent the power supply. The motor will operate if the start

button A is pressed and if the overload contact in the motor circuit is closed.

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FIG. 3

In this case, A contacts would normally be open and B contacts closed. The

statement describing the circuit, assuming the supply is connected would be:

Current will flow to the motor if A contacts close and B contacts do not open.

F = A AND NOT B

i.e.

Such diagrams are used to show sets of normally open or normally closed

contacts, as for example in FIGURE 4. In this case R is a relay that will be

activated by the closing (or not opening) of the contacts. Contact R3 is

normally closed, the rest are normally open.

FIG. 4

R1 R3 R4

F A B= .

+ve Start MotorOverload contact

–ve

A B M

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Write down the Boolean equation for the statement describing the operation of relay R

in FIGURE 4.

....................................................................................................................................................

________________________________________________________________________________________

FIGURE 4, loosely resembling the rung of a ladder, could be part of a diagram

consisting of several ‘rungs’. Such diagrams are referred to as ‘Ladder

Diagrams’ and we shall be meeting them again later in the module.

F R R R R= +( ) . .1 2 3 4

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________________________________________________________________________________________

USING TRUTH TABLES TO DETERMINE EQUIVALENT BOOLEAN EXPRESSIONS ________________________________________________________________________________________

Truth tables can be used to determine whether two ostensibly different Boolean

expressions are in fact the same or equivalent. If the truth tables for two

different expressions have the same outcome or dependent variable values,

then the expressions are identical, even though their variables and operators

may be different.

A + B does not look much like , but in fact they are different forms of

the same relationship between the variables.

Example

Show by compiling truth tables that:

We have already encountered the truth table for A + B, shown here again in

TABLE 4:

TABLE 4

F = A + B

A B A B+ = .

A B.

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TABLE 5 shows truth table for

TABLE 5

It can be seen that the two expressions have the same outcome for the same

combination of values of the variables, so the expressions are regarded as

identical and could be interchanged.

Show by means of truth tables that:

________________________________________________________________________________________

A A . B A B+ = +

F = A . B

A . B

A B. :

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The truth tables are shown in TABLES 6 and 7.

TABLE 6

TABLE 7

The truth tables are the same, therefore the functions have the same values.

Note: (i) When dealing with several variables, we must ensure that all

combinations of 0's and 1's are contained in the table. To

facilitate this we use the binary number system.

F = A + B

F = A + A . B

A . B

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When dealing with whole binary numbers, 20 is often termed the Least

Significant Bit (LSB) where the binary number alternates between 0 and 1.

The next column of 0’s and 1’s corresponding to the next higher power of 2,

has changes every two binary numbers, the next column every four binary

numbers and so on. The highest power of 2 is termed the Most Significant Bit

(MSB). Adopting this pattern ensures all combinations of 1’s and 0’s for all

variables are covered.

We shall encounter number systems again later in the lesson where we will

meet number systems in bases other than 10 and 2.

(ii) Two tables have been compiled and A appears in both. It is

quicker and more convenient to compile only one truth table

containing (in this case) both expressions, in this way,

duplication of variable listing is avoided.

0 1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1

0 0 0 0 1 1 1 1 0

0 0 1 1 0 0 1 1 0

0 1 0 1 0 1 0 1 0

D C B A

23 22 21 2024Denary number

Binary number

Change after every one Change after every two Change after every four Change after every eight

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________________________________________________________________________________________

SIMPLIFICATION OF BOOLEAN EXPRESSIONS ________________________________________________________________________________________

SIMPLIFICATION USING TRUTH TABLES

(For convenience, we shall dispense with the AND operator when writing

expressions except where inclusion will increase clarity.)

Truth tables can also be used as an aid when simplifying Boolean expressions.

If we examine the expression:

it looks quite complicated. If we construct the truth table (shown in TABLE 8)

we can see that the expression can be simplified.

TABLE 8

We can see that the output F is only 1 when the input variables A and B equal 0

and 0 respectively, that is:

F A B=

F = A B (AB + B)

AB + B

A B

F A B AB B= +( )

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The truth table for confirms this (column six of the above truth table).

We have then,

Often, by inspection of a truth table, it is possible to obtain a simplified

expression. If we construct the truth table for the expression:

TABLE 9 shows that we shall require eight combinations of variable values in

the truth table (note the order of the variables is not important as long as all

combinations are covered):

TABLE 9

ABC

A B C

ABC

F A B C A B C A B C A B C= + + +

F A B AB B A B= + =( )

A B

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In this case nothing is gained by writing down the values of the input variables

on the occasions when F = 1 as one is left with the same expression. By

inspection however, we see that the F column is identical with the input

variable B column so:

F A B C A B C A B C A B C B= + + + =

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________________________________________________________________________________________

THE REPRESENTATION OF LOGICAL FUNCTIONS USING ELECTRONIC GATES ________________________________________________________________________________________

LOGIC CIRCUITS

Boolean logic can be physically represented by electromechanical and fluid

operated switches and relays. Logic circuits now are, almost universally,

circuits comprising electronic gates.

Electronic gates are devices which perform a particular Boolean logic function.

They can have several inputs and only one output. Inputs and outputs can only

assume one of two states which are usually electronically represented as two

voltage levels, one low, designated '0', and one high, designated ‘1’.

(Note: There exist what are known as ‘tri-state’ logic gates whose construction

allows a third state of effectively ‘not connected’ for computer hardware

purposes, but which has no effect on the gate's Boolean operation).

STANDARD GATE SYMBOLS

Two sets of gate symbols representing Boolean functions are currently in use:

• British Standards Institute (BSI), publication BS 3939

• American National Standards Institute (ANSI) publication Y32.14 – 1973.

The latter are almost universally used and will be used in this module.

FIGURE 5 shows the OR, AND and NOT gate symbols in the ANSI system.

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FIG. 5

We should note from FIGURE 5 that:

(i) three-input OR and AND gates are shown though the number of

inputs may be more

(ii) the symbol for the NOT or inverter gate is the symbol for an

electronic amplifier with a small circle at the output. The small

circle indicates an inversion or the opposite of the input

variable. For example, if the input of the NOT gate is a ‘1’ then

the output will be a ‘0’ and vice versa. On other gates, for

example NAND, NOR and EXNOR gates the circle on the

output of the gate indicates inversion of the gate function and

not inversion of the gate inputs.

NOR gate

This gate can be represented as an OR gate whose output is connected to the

input of a NOT gate as shown in FIGURE 6(a).

OR gate: A B C

A B C

F = A + B + C

F = A . B . C

F = A

AND gate:

NOT gate:

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FIG. 6(a)

This combination is represented in the ANSI system as that shown in FIGURE 6(b).

FIGURE 6(c) shows the truth table for the gate.

FIG. 6(b) FIG. 6(c)

F = A + B + C A B C

A B C

F = A + B + C

A + B + C

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Note: A NOR gate with inverted inputs, or active low inputs, behaves like an

AND gate, that is, if all inputs are 0, then the output is 1. This is known as a

Negative AND gate. FIGURE 7 shows the ANSI symbol.

FIG. 7

NAND gate

This gate can be represented as an AND gate whose output is connected to the

input of a NOT gate as shown in FIGURE 8(a).

FIG. 8(a)

This combination is represented in the ANSI system as that shown in FIGURE 8(b).

FIGURE 8(c) shows the truth table for the gate.

A B C

F = ABC

ABC

A B C

F = A . B . C

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FIG. 8(b) FIG. 8(c)

EXCLUSIVE OR (or XOR) gates can be made from combinations of AND,

OR and NOT gates, as shown in FIGURE 9(a). For convenience, only two-

input gates are shown, though more inputs can be used. FIGURE 9(b) shows

the truth table for the gate.

FIG. 9(a) FIG. 9(b)

F = AB + AB

A B C

F = ABC

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FIGURE 10 shows the US (ANSI) and British symbols for the main types of

gate.

FIG. 10

AND

NOR

NAND

Exclusive OR

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________________________________________________________________________________________

SELF-ASSESSMENT QUESTIONS ________________________________________________________________________________________

1. Draw the logic circuit to represent the Boolean expression

2. Write down the Boolean expression for the output F of FIGURE 11.

Insert how the expression is obtained on the diagram

FIG. 11

3. Compile a truth table to show that C ABC C( ) = .

C F

F AC D AB CD BC= + +( )

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________________________________________________________________________________________

NOTES ________________________________________________________________________________________

...................................................................................................................................................

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________________________________________________________________________________________

ANSWERS TO SELF-ASSESSMENT QUESTIONS ________________________________________________________________________________________

1. The logic circuit and how the expression is derived is shown in

FIGURE 12.

where

FIG. 12

FIG. 11 Reproduced with gate outputs shown

C F

BC C

B ABC

The output expression is F ABC AC BC= + +

F AC D AB CD BC= + +( )

ACD

(CD + BC) AB (CD + BC)

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3. The truth table is shown in TABLE 10:

TABLE 10

From which it can be seen that the column for F is the same as that for

A B C

F = C (A B C)

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________________________________________________________________________________________

SUMMARY ________________________________________________________________________________________

In this lesson we have encountered Boolean operators and functions. We have

also encountered truth tables which can be used to simplify Boolean

expressions. Logic circuits using standard logic gate symbols can be drawn for

such Boolean expressions. Although other minimisation techniques exist for

the manipulation of Boolean expressions, they are not immediately relevant to

programmable logic controllers and are therefore not covered in this module.

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setdistillerparams << /HWResolution [2400 2400] /PageSize [612.000 792.000] >> setpagedevice