Lecture 2

2. Number systems, codes, signals - fundamentals2.1 The decimal number system

2.2 The binary number system2.3 The BCD code 2.4 The hexadecimal number system2.5 Signed binary numbers2.6 Real numbers 2.7 Generation of binary and digital signals

3. Boolean operations - fundamentals3.1 Basic logic functions3.2 Further logic operations3.3 Establishing switching functions3.4 Simplification of logic functions3.5 Karnaugh-Veitch diagram

2.1 The decimal number system

• Characteristic of the decimal number system used in general is the linear array of digits and their significant placing.

Example:The number 4344, for instance, can berepresented as follows:4344 = 4 x 1000 + 3 x 100 + 4 x 10 + 4x1

Number 4 on the far left is of different significance to that of number 4 on the far right.

The decimal number system - 2• The basis is 10.• 10 different digits permit counting from 0

to 9.• If counting is to exceed the number 9, this

constitutes a carry over to the next place digit.• The significance of this place is 10, and the

next carry over takes place when 99 is reached etc.

„Decimal“: originating from the latin 'decem' = 10

The decimal number system - 3

• Weights of particular places are powers of 10• 718 711=

7x100000+1x10000+8x1000+7x100+1x10+1x1

• The digit on the far right is referred to as the least significant digit (1), and the digit on the far left as the most significant digit (7).

The decimal number system - 4

• Any number system can be configured on the basis of this example.

• The fundamental structure can be applied to number systems of any number of digits.

• Consequently, any computing operations and computing methods which use the decimal number system can be applied with other number systems.

2.2 The binary number system

• We are indebted to Leibnitz, who applied the structures of the decimal number system to two-digit calculation.

• As long ago as 1679, this created the premises essential for the development of the computer, since electrical voltage or electrical current only permits a calculation using just two values: e.g. "current on", "current off". These two values are represented in the form of digits: "1" and „0".

The binary number system - 2• If one were to be limited to exactly 2 digits per

place a number system would be configured as follows:

• Weights are the powers of the digit „2“:128, 64,32,….

Because of the exclusive use of two digits, this number system is known as the binary or also the dual number system

The binary number system - 3

Example ( 8 places ):

Value of the number (8 places):1x128+0x64+1x32+1x16+0x8+0x4+0x2+1x1=177

The individual places of the binary number system can adopt one of the two digits 0 or 1

1 0 1 1 0 0 0 1

The binary number system - 4

• Up to a maximum of (2 powered by 8 – 1) = 256 - 1 = 255 can be calculated with eight places, which would be the number 1111 1111.

• 1 bit 1 digit • 1 nibble 4 digits ( a half of a byte)• 1 byte 8 digits• 1 word 16 digits• 1double word 32 digits

The binary number system - 5

• Generally up to a maximum of

(2 powered by n – 1)

can be expressed with n places.

Example:

1.How many various values can we obtain using 10 places A/D converter??

2.What is the maximum accuracy of the convertor?

The binary number system - 6

Ad 1:(2 powered by n – 1)=[n=10]=(2 powered by 10 – 1)=1024-1=1023

Maximum value of 10 digit binary number is 1023.

Ad 2:Maximum accuracy of the convertor 1/1023=approximatelly 1 promille

The binary number system - 7

• A/D converter:

Analog valueEg. (0-10) V 4 V

Digital value (binary number)Eg. (0-1023) 1023/10 x 4=(409,2)=409

2.3 The BCD code• For people used to dealing with the decimal

system, binary numbers are difficult to read.• For this reason , a more easily readable numeral

representation was introduced, i.e. the binary coded decimal notation, the so-called BCD code (Binary Coded Decimal).

• With this BCD code, each individual digit of the decimal number system is represented by a corresponding binary number

The BCD code - 2

• Representation of decimal numbers in BCD code

The BCD code - 3

• 4 digits in binary notation are therefore required for each digit in the decimal system

Example:

Decimal number 7133 is thus represented

0111 0001 0011 0011

in this way in BCD code

The BCD code - 4

Example:

Number 0101 0010 0111 in BCD code

represents 527 decimal number

BCD coded numbers are often used for seven segment displays and coding switches

2.4 The hexadecimal number system

• The use of binary numbers is often difficult (long, not providing an easy survey)

• The BCD code takes up a lot of space

• This is why the octal (3 bits,8 various digits: 0-7) and hexadecimal (4 bits, 16 various digits: 0-F) systems were developed

The hexadecimal number system-2

• Usable digits at one place of the number

0 1 2 3 4 5 6 7 8 9 A B C D E F

(0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)

• Base is the number 16

• Weights are the powers of the digit „16“:

1,16,256,4096….

The hexadecimal number system-3

• Example:

Hexadecimal number 87BC has in decimal system value 34 748

8x4096+7x256+11x16+12x1=34 748

The hexadecimal number system-4• Often is convenient to express long binary

number in hexadecimal system for bettter readability

Example:Double word binary number:

0110010111111001Groups of 4 binary digits

0110 0101 1111 1001Number in hexadecimal system

6 5 F 9

The hexadecimal number system-5

Example:Number in hexadecimal system

4 D 5 Groups of 4 binary digits

0100 1110 0101 Double word binary number:

010011100101 0000 010011100101

2.5 Signed binary numbers

• Up to now, we have dealt solely with whole positive numbers, not taking into account negative numbers.

• To enable working with these negative numbers, it was decided that the most significant bit on the far left of a binary number is to be used to represent the preceding sign: – „0" thus corresponds to "+" and – "1" corresponds to „-".

Signed binary numbers -2

Example:

Signed binary number

11111111 represents -127 decimal number

Signed binary number

01111111 represents +128 decimal number

On one byte (8 digits) we can express signed numbers in range (-127 to +128 )

Signed binary numbers -3

• Since the most significant bit has been used, one bit less is available for the representation of a signed number.

• One byte (8 binary digits)

Integer Range of values

Unsigned 0 to 255

Signed -127 to + 128

Signed binary numbers -4

• Two bytes – one word (16 binary digits)

Integer Range of values

Unsigned 0 to 65 535

Signed -32768 to + 32767

2.6 Real numbers

• Although it is now possible for whole positive and whole signed numbers to be represented with 0 or 1 , there is still the need for points or real numbers.

• In order to represent a real number in computer binary notation, the number is split into two groups, a power of ten and a multiplication factor. This is also known as the scientific representation of digits.

Real numbers - 2

• The number 27,3341 is thus converted into 273 341 x 10 powered by -4.

• Two whole signed numbers are therefore required for a real number to be repre- sented in a computer or PLC.

2.7 Generation of binary and digital signals

• As has already become clearly apparent in the previous section, all computers and as such all PLCs operate using binary or digital signals.

• By binary signal, we understand a signal which recognises only two defined values

• These values are termed „0" or "1", the terms „Iow“ and „high“, or abreviations „L“ and „H“ are also used.

Generation of binary and digital signals - 2

• Binary signal

1

0

t

Generation of binary and digital signals - 3

• The signals can be very easily realised with contacting components.

-an actuated normally open contact (NOC) corresponds to a logic 1-signal and

-an unactuated one to a logic 0-signal.• When working with contactless components, this

can give rise to certain tolerance bands. For this reason, certain voltage ranges have been defined as logic 0 or logic 1 ranges.

Generation of binary and digital signals - 4

• When working with contactless components, this can give rise to certain tolerance bands. For this reason, certain voltage ranges have been defined as logic 0 or logic 1 ranges.

Generation of binary and digital signals - 5

• Voltage ranges

30

11

5

0

-3

V

1 range

0 range

Generation of binary and digital signals - 6

• lEC 1131-2 defines a value range of -3 V to 5 V as logic 0-signal, and

11 V to 30 V as logic 1-signal (for contactless sensors).This is binding for PLCs, whose device technology is

to conform to lEC 1131-2.

In current practice, however, other voltage ranges can often be found for logic 0- and 1-signal. Widely used are: -30 V to +5 V as logic 0,

+ 13 V to 30 V as logic 1.

Generation of binary and digital signals - 7

• Unlike binary signals, digital signals can assume any value.

• These are also referred to as value stages.

• A digital signal is thus defined by any number of value stages. The change between these is non-sequential.

• Any???, How many??? Example

Generation of binary and digital signals - 8

• The following illustration shows three methods of converting an analogue signal into a digital signal, depending on the step hight set.

• We use step hight of– 3 V– 1 V– 0.5 V

Generation of binary and digital signals - 9

step hight 3V

1

2

3

4

5

V

t

0

Generation of binary and digital signals - 10

step hight 1V

1

2

3

4

5

V

t0

Generation of binary and digital signals - 11

step hight 0,5V ????

1

2

3

4

5

V

t0

Generation of binary and digital signals - 12

• Digital signals may be formed from analogue signals. This method is for instance used for analogue processing via PLC. Accordingly, the analogue input signal within a range of 0 to 10 V is reduced into a series of step values.

• Depending on the quality of the PLC and the possible step height set, the digital signal would thus be able to operate in steps of value of 0.1 V, 0.01 V or 0.001 V.

• Naturally, the smallest possible range is to be selected in order for the analogue signal to be reproduced as accurately as possible.

Generation of binary and digital signals - 13

• Simple example of an analogue signal is pressure, which is measured and displayed by a pressure gauge. The pressure signal may assume any intermediate value between its minimum and maximum values. Unlike the digital signal, it changes continually. In the case of the processing of analogue values via a PLC, as described, analogue voltage signals are evaluated and converted.

Generation of binary and digital signals - 14

Analog value

Analog input modul

CPU Output modul

A/D convertor

Digital value

(1 byte)(Pressure p)

PLC

Generation of binary and digital signals - 15

Digital values on one byte - 8 bits

Generation of binary and digital signals - 16

• On the other hand, digital signals can be formed by adding together a certain number of binary signals.

• In this way again, as described above, it is also possible to generate digital signal with 256 (8 bits), 65 535 (16 bits) values.

• This is for instance used to implement timer and counter functions.

Literature

• Nripendra N. Biswas: Logic Design Theory,Prentice Hall International,1993,ISBN 0-13-010695-X