EE088-Digital Technique 1-Th-Inst.pdf

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SRI LANKA INSTITUTE of ADVANCED TECHNOLOGICAL EDUCATION

Training Unit

Digital Technique 1Theory

No: EE 088

INDUSTRIETECHNIKINDUSTRIETECHNIK

ELECTRICAL and ELECTRONIC

ENGINEERING

Instructor Manual


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Training Unit

Digital Technique 1

Theoretical Part

No.: EE 088

Edition: 2008Al l Rights Reserved

Editor: MCE Industrietechnik Linz GmbH & CoEducation and Training Systems, DM-1Lunzerst rasse 64 P.O.Box 36, A 4031 Linz / Aus triaTel. (+ 43 / 732) 6987 - 3475Fax (+ 43 / 732) 6980 - 4271Website: www.mcelinz.com


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DIGITAL TECHNIQUE 1

CONTENTS Page

LEARNING OBJECTIVES...................................................................................................3

1 INTRODUCTION ..........................................................................................................4

1.1 Structure of the Electronics..................................................................................4

1.2 Types of signals ...................................................................................................5

1.2.1

Analogue signals..............................................................................................5

1.2.2 Digital signals ...................................................................................................5

1.3 Bit.........................................................................................................................6

1.3.1

Binary digital.....................................................................................................6

2

NUMBER SYSTEMS....................................................................................................7

2.1 Representation of numbers..................................................................................7

2.2 Decimal System ...................................................................................................8

2.2.1 Decimal System ...............................................................................................9

2.2.2 System of five:..................................................................................................9

2.2.3 Octal system ..................................................................................................10

2.2.4 Hexadecimal system: .....................................................................................10

2.2.5 Dual system ...................................................................................................11

2.2.6

Dual numbers from 0 to 16.............................................................................12

2.2.7 Converting a decimal into a dual >1...............................................................13

2.2.8

Conversion of a decimal into a dual 1...................................................16

2.2.10 Converting duals into decimals < 1 ............................................................18

3 CALCULATING IN THE DUAL SYSTEM ...................................................................20

3.1

Addition ..............................................................................................................20

3.2 Multiplication ...................................................................................................... 25

3.3

Subtraction.........................................................................................................31

3.4 Division ..............................................................................................................34

3.5

Subtracting by adding the complement of the subtrahend.................................36

3.5.1 Decimal system..............................................................................................36

3.5.2 Dual system ...................................................................................................37

3.6

One's complement .............................................................................................38


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DIGITAL TECHNIQUE 1

LEARNING OBJECTIVES

The trainee should be able to...

distinguish between analogue signals and digital signals

allocate digits to defined quantities

explain the basic principle of number systems

explain the principle of the binary representation of numbers

give examples for the principle of the octal and hexadecimal representation of

numbers

convert every decimal number into a binary number

perform the four fundamental operations using the binary system subtract in the binary

system using the one's complement.


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DIGITAL TECHNIQUE 1

1 INTRODUCTION

1.1 Structure of the Electronics

The digital technology includes automation technology which deals with the design and

the construction of control systems where information is being transmitted by means of

digital signals.

The term "digital" is derived from the Latin word "digitus" which means "finger or

"number. Since people formerly just used their fingers for counting, the term "digital"

simply means "related to figures".

NOTE:

In digital technology, all signals must therefore be understood as number!


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1.2 Types of signals

1.2.1 Analogue signals

Analogue signals may take any intermediate values within their technical limits (range).

Examples:

The voltage supplied by a tacho-generator is directly proportional to the number of

revolutions.

The length of the mercury column of a thermometer is directly proportional to the

temperature.

The voltage emitted by a thermo-couple is directly proportional to the temperature.

1.2.2 Digital signals

Digital technology operates almost exclusively with binary signals. Only two different

values may be assumed ("0" or "1").


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Examples:

The contact of a switch may only be open or closed.

A bulb may only be switched on or switched off.

If an analogue value should be represented as a digital value it must be quantized in

accordance with a fundamental unit. Here, an equivalent stepwise course is allocated to

the continual course of the value to be signalled.

1.3 Bit

1.3.1 Binary digital

The signal change from 0 -> 1 or from 1 -> 0 contains the information (Information

quantity) of 1 bit.


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2 NUMBER SYSTEMS

2.1 Representation of numbers

Task:

Draw circles around identical quantities and allocate a defined symbol to every quantity.

Solution:

Different quantities are expressed by different "numbers".

A digit is the symbol for a defined quantity. Yet, since a quantity may be of any desired

value one would need also any desired number of symbols (digits) for it to be clearly

recognized.

This will not be necessary as soon as larger quantities are systematically bundled. Digits

which represent quantitative bundles must be specially identified to allow their

differentiation. This is done by the spatial position of the digit.


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2.2 Decimal System

Task:

Bundle this quantity to packages of ten, allocate symbols and determine the total quantity:

Solution:

In everyday life, the decimal system is used. It is based on the 10. In every position of a

number ten different conditions are possible, namely, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and

9. Additionally, each position is allocated to a value (weight), for example, units, tens,

hundreds, etc.


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2.2.1 Decimal System

Basis 10

Numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

Example:

NOTE:

In the decimal system, the total quantity is calculated from the sum of the products of

digits and positional values.

For the representation of numbers, other basic digits may be used as well. The procedure

is the same as for the decimal system.

2.2.2 System of five:

Basis 5

Numbers (0, 1, 2, 3, 4)


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Example:

2.2.3 Octal system

Basis 8

Numbers (0, 1, 2, 3, 4, 5, 6, 7)

Example:

2.2.4 Hexadecimal system:

Basis 16

Combinations of numbers and letters (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F; in order

to avoid 2-digit numerals above 9, the hexadecimal numbers from 10 to 15 are identified

by the letters A to F). The hexadecimal system is also often called sedecimal system and

is frequently used in computer technology.


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Example:

Conversion of the decimal number 50345 into a hexadecimal number:

2.2.5 Dual system

(Latin "duo" = 2)

In the dual system, the Basis digit is 2. Only the binary digits 0 and 1 are needed. It is the

ideal system to be used in electronic equipment as there are only two logical states to

which the two possible digits are allocated.

Example:

The configuration of a multi-digit, number is done according to the same laws that govern

the decimal system.


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2.2.6 Dual numbers from 0 to 16

Task:

Continue the dual numbers until 49:

Solution:

There are procedures which are used to convert decimals Info dual numbers, and vice

versa.


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2.2.7 Converting a decimal into a dual >1

The decimal is continually divided by 2 (division procedure). The rest (0 or 1), read from

left to right, is the dual number.

Decimal 47

Examples:

1. Decimal 53

2. Decimal 56


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3. Decimal 84 =

4. Decimal 103 =

5. Decimal 128 =

2.2.8 Conversion of a decimal into a dual 1 is expressed by "1" which is then always subtracted from the product.

Decimal 0.65625 =


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Examples:

1. Decimal 0,321 =

2. Decimal 0,125 =

3. Decimal 0,034 =

4. Decimal 0,5 =

5. Decimal 0,25 =


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2.2.9 Converting dual numbers into decimals > 1

The conversion procedure for decimals into dual is inverted. Instead of dividing by 2 it is

now multiplied by 2. Starting from left, the first digit of the dual number is multiplied by 2,

then the next digit (0 or 1) is added, and the total again multiplied by 2.

Examples:

1.

2.


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3.

4.

5.

6.


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2.2.10 Converting duals into decimals < 1

The dual digits behind the comma are continually divided by 2, starting from right to left.

The dual digit before the comma is always added.

Dual 0,10101 =

Examples:

1. Dual 0.001 =

2. Dual 0.C1101 001


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3. Dual 0,01

4. Dual 0,01011

5. Dual 0,1011


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3 CALCULATING IN THE DUAL SYSTEM

Calculating in the dual system follows the same rules as for the decimal system. As there

are only two dual digits these rules are particularly simple. The tour fundamental

operations can be carried out on the basis of one addition procedure.

3.1 Addition

The following rules apply within the individual digits:


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Examples:

1.

2.


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3.

4.


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5.

6.


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7.

8.


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3.2 Multiplication

The following rule applies:


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It happens quite frequently that several units (1) have to be added to form a sum. This will

result in several units (1) to be carried over to the next higher dual number.

Carries during summarisation:

Examples:

1.


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2.

3.


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4.

5.


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6.

Especially for larger multiplications it may become difficult to perform by straight binary

coding addition.

The procedure is facilitated if a partial product is formed.


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Example:

7.


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3.3 Subtraction

The following rule applies: Minuend - subtrahend


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Examples:

1.

2.


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3.

4.


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3.4 Division

Divisions are performed on the basis of individual subtractions. The divisor is subtracted

from a digit of the dividend which must be great enough to give 0 as a result. The next

lower digit of the dividend is then pulled down to the resulting difference and divides again

by the divisor.

Examples:

1.

Dividend: divisor quotient


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2.

3.

4.


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5.

3.5 Subtracting by adding the complement of the subtrahend

With the help of complements, subtraction can be converted to an addition. Consequently,

data processing units need only an electronic adder for computing.

NOTE:

The complement of a number is the supplement of this number to any other desired

number.

3.5.1 Decimal system

Nines complement:Here, every decimal digit is completed to the next high decimal digit 9 - which is called a

"nines complement".


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As the nines complement is always less by 1 than the tenths complement, that 1 must be

added after the addition of the complement in order to obtain a correct result.

Nines complement of 17 = 82 Nines complement of 163 = 836

3.5.2 Dual system

In the dual system, the procedure is as follows:

As there are only 2 dual digits for each dual number (0, 1), completion towards the highest

dual digit is as simple as it can be.

For 0 -> 1 (0 + 1 = 1)

For 1->a0 (1 + 0 = 1)

NOTE:

Only one inversion needs to be carried out. Analogously to the nines complement this is

called "ONE'S COMPLEMENT".


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3.6 One's complement

The one's complement of a binary digit is obtained by negating the individual dual

numbers


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EE088 - Digital Technique 1

Theoretical Test


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DIGITAL TECHNIQUE 1

TEST 1

1. Explain the term 'total quantity in various number systems.

2. Convert the following hexadecimal number into a decimal number. 12FD1/16 =

3. Convert the following octal number into a decimal number. 6405/8 =

4. Convert the decimal number 156.33 into a dual number with four decimals after the

comma.

5. Convert the dual number 1000111, 0110011 into a dual number with four decimals

behind the comma.

6. Add the following dual number and write the carries:

1100101

111111

7. Give the subtraction rules for the dual system.


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8. Multiply the following binary number and write the carries when making the additions.

1011 . 1101

9. Perform the following division in the dual system:

1001101 : 111 =

10. Subtract the following binary numbers using the one's complement:

110111

- 1000

11. Explain the term 1 bit.

12. Convert the following hexadecimal number into a decimal number:

BCD

13. Convert the following octal number into a decimal number:

1750/8 =

14. Convert the decimal number 253.77 into a binary number with 4 decimals behind the

comma.

15. Convert the binary number 101100.101 into a decimal number with 3 decimals behind

the comma.

16. Add the following binary number and write the carries:

11011011

+ 10111101

17. Subtract the following binary number and write the borrowers:

1111001

- 1011101


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18. Multiply the following binary number and write the carries when making the addition:

1101 . 1011

19. Perform the following division in the dual system:

1011000: 1000 =

20. Subtract the following binary number using the one's complement:

111010

- 10000


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DIGITAL TECHNIQUE 1

TEST 1

(Solution)

1. It is calculated from the sum of the products of digits and positional values.

2.

3.

4.

5.


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6.

7.

8.

9.

10.


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11. Measuring unit for an information.

The signal change from 0 to 1 and from 1 to 0.

12.

13.

14.

15.


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16.

17.

18.

19.

20.


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