FUNDAMENTALS OF DIGITAL CIRCUITS by anand kumar - PHI Learning

The Fourth edition of this well-received text continues to provide coherent and comprehensive coverage of digital circuits. It is designed for the undergraduate students pursuing courses in areas of engineering disciplines such as Electrical and Electronics, Electronics and Communication, Electronics and Instrumentation, Telecommunications, Medical Electronics, Computer Science and Engineering, Electronics, and Computers and Information Technology. It is also useful as a text for MCA, M.Sc. (Electronics) and M.Sc. (Computer Science) students. Appropriate for self study, the book is useful even for AMIE and grad IETE students.

Written in a student-friendly style, the book provides an excellent introduction to digital concepts and basic design techniques of digital circuits. It discusses Boolean algebra concepts and their application to digital circuitry, and elaborates on both combinational and sequential circuits. It provides numerous fully worked-out, laboratory tested examples to give students a solid grounding in the related design concepts. It includes a number of short questions with answers, review questions, fill in the blanks with answers, multiple choice questions with answers and exercise problems at the end of each chapter.

As the book requires only an elementary knowledge of electronics to understand most of the topics, it can also serve as a textbook for the students of polytechnics, B.Sc. (Electronics) and B.Sc. (Computer Science).

1. Fundamentals of Digital Circuits Fourth Edition A. ANAND KUMAR Advisor to Dean Academics and Dean Quality K.L. University Green Fields, Vaddeswaram Guntur District Andhra Pradesh © PHI Learning Private Limited

2. Chapter 2 NUMBER SYSTEMS

3. NUMBER SYSTEMS Types of Number System Decimal number system Binary number system Octal number system Hexadecimal number system

4. THE DECIMAL NUMBER SYSTEM Contains ten unique symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 Since counting in decimal involves ten symbols, we say that its base or radix is ten .

5. SUBTRACTION OF DECIMAL NUMBERS Subtraction of decimal numbers can be accomplished by the 9’s and 10’s complement methods . 9’s Complement Method of Subtraction To perform decimal subtraction using the 9’s complement method, obtain the 9’s complement of the subtrahend and add it to the minuend. 10’s Complement Method of Subtraction To perform decimal subtraction using the 10’s complement method, obtain the 10’s complement of the subtrahend and add it to the minuend.

6. THE BINARY NUMBER SYSTEM The binary number system is a positional weighted system. The base or radix of this number system is 2 . The symbols used are 0 and 1. A binary digit i.e. 0 or 1, is called a bit . Counting in Binary Counting in binary is very much similar to decimal counting as shown in Table.

7. THE BINARY NUMBER SYSTEM ( Contd .)

8. BINARY TO DECIMAL CONVERSION Binary numbers may be converted to their decimal equivalents by the positional weights method. In this method, each binary digit of the number is multiplied by its position weight and the product terms are added to obtain the decimal number.

9. DECIMAL TO BINARY CONVERSION There are two methods to convert a decimal number to a binary number. These are the reverse processes of the two methods used to convert a binary number to a decimal number. First method In this method, which is normally used only for small numbers, the values of various powers of 2 need to be remembered . Second method In this method, the decimal integer number is converted to the binary integer number by successive division by 2, and the decimal fraction is converted to binary fraction by successive multiplication by 2. This is also known as the double-dabble method.

10. RULES FOR BINARY ARITHMETIC OPERATIONS Binary Addition The rules for binary addition are the following: 0 + 0 = 0; 0 + 1 = 1; 1 + 0 = 1; 1 + 1 = 10, i.e. 0 with a carry of 1 . Binary Subtraction The rules for binary subtraction are: 0 – 0 = 0; 1 – 1 = 0; 1 – 0 = 1; 0 – 1 = 1, with a borrow of 1 . Binary Multiplication There are two methods of binary multiplication—the paper method and the computer method.

11. RULES FOR BINARY ARITHMETIC OPERATIONS Computer Method of Multiplication A computer can add only two numbers at a time with a carry. So, the paper method cannot be used by the digital computer

12. RULES FOR BINARY ARITHMETIC OPERATIONS Binary Division Like multiplication, division too can be performed by two methods—the paper method and the computer method . Computer Method of Division The computer method of division involves successive subtraction.

13. 2’s COMPLEMENT OF A NUMBER Methods of obtaining the 2’s complement of a number: The 2’s complement of a number can be obtained in three ways as given below. By obtaining the 1’s complement of the given number (by changing all 0s to 1s and 1s to 0s) and then adding 1. By subtracting the given n-bit number N from 2n. Starting at the LSB, copying down each bit up to and including the first 1 bit encountered, and complementing the remaining bits.

14. 1’s COMPLEMENT OF A NUMBER Method of obtaining the 1’s complement of a number : The 1’s complement of a number is obtained by simply complementing each bit of the number, that is, by changing all the 0s to 1s and all the 1s to 0s.

15. THE OCTAL NUMBER SYSTEM The octal number system is also a positional weighted system. Its base or radix is 8. It has 8 independent symbols 0, 1, 2, 3, 4, 5, 6, and 7. Since its base 8 = 23, every 3-bit group of binary can be represented by an octal digit. An octal number is, thus, 1/3rd the length of the corresponding binary number. Usefulness of the Octal System When dealing with large binary numbers of many bits, it is convenient and more efficient for us to write the numbers in octal rather than binary. However, the digital circuits and systems work strictly in binary; we use octal only for the convenience of the operators of the system.

16. OCTAL COUNTING

17. OCTAL TO BINARY CONVERSION AND VICE VERSA Octal to Binary Conversion To convert a given octal number to a binary, just replace each octal digit by its 3-bit binary equivalent . Binary to Octal Conversion To convert a binary number to an octal number, starting from the binary point make groups of 3 bits each, on either side of the binary point, and replace each 3-bit binary group by the equivalent octal digit .

18. OCTAL TO DECIMAL CONVERSION Octal to Decimal Conversion To convert an octal number to a decimal number, multiply each digit in the octal number by the weight of its position and add all the product terms . Decimal to Octal Conversion To convert the given decimal integer number to octal, successively divide the given number by 8 till the quotient is 0. The last remainder is the MSD. The remainders read upwards give the equivalent octal integer number. To convert the given decimal fraction to octal, successively multiply the decimal fraction and the subsequent decimal fractions by 8 till the product is 0 or till the required accuracy is obtained. The first integer from the top is the MSD. The integers to the left of the octal point read downwards give the octal fraction.

19. THE HEXADECIMAL NUMBER SYSTEM Binary numbers are long. These numbers are fine for machines but are too lengthy to be handled by human beings. So, there is a need to represent the binary numbers concisely. One number system developed with this objective is the hexadecimal number system (or Hex ). The hexadecimal number system is a positional-weighted system. The base or radix of this number system is 16, that means, it has 16 independent symbols. The symbols used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Since its base is 16 = 24, every 4 binary digit combination can be represented by one hexadecimal digit. A 4-bit group is called a nibble. It is 1/4th the length of the corresponding binary number, yet it provides the same information as the binary number .

20. HEXADECIMAL CONVERSIONS Binary to Hexadecimal Conversion Hexadecimal to Binary Conversion Hexadecimal to Decimal Conversion Decimal to Hexadecimal Conversion Octal to Hexadecimal Conversion Hexadecimal to Octal Conversion

21. HEXADECIMAL CONVERSIONS Binary to Hexadecimal Conversion To convert a binary number to a hexadecimal number, starting from the binary point, make groups of 4 bits each, on either side of the binary point and replace each 4-bit group by the equivalent hexadecimal digit as shown in Table.

22. Binary to Hexadecimal Conversion HEXADECIMAL CONVERSIONS ( Contd .)

23. HEXADECIMAL CONVERSIONS Hexadecimal to Binary Conversion To convert a hexadecimal number to binary, replace each hex digit by its 4-bit binary group . Hexadecimal to Decimal Conversion To convert a hexadecimal number to decimal, multiply each digit in the hex number by its position weight and add all those product terms . Decimal to Hexadecimal Conversion To convert a decimal integer number to hexadecimal, successively divide the given decimal number by 16 till the quotient is zero. The last remainder is the MSB. The remainders read from bottom to top give the equivalent hexadecimal integer .

24. HEXADECIMAL CONVERSIONS ( Contd .) Octal to Hexadecimal Conversion To convert an octal number to hexadecimal, the simplest way is to first convert the given octal number to binary and then the binary number to hexadecimal . Hexadecimal to Octal Conversion To convert a hexadecimal number to octal, the simplest way is to first convert the given hexadecimal number to binary and then the binary number to octal.

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