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1. SECA1201
DIGITAL LOGIC CIRCUITS
Unit –1

2. Number Systems
Number System is a basis for counting various
items
•Decimal Number System (0 –9)
•Binary Number System (0, 1)
•Octal Number System ( 0 –7)
•HexaDecimal Number System (0 –9, A –F)

3. Number –Base –Digits
Number Systems

4. Counting of Number Systems
Decimal Counting
0,1,2,3,4,5,6,7,8,9,10,11,12,13.........
Binary Counting
1 bit binary counting ( Total count 2
1
= 2)
2 bit binary counting ( Total count 2
2
= 4)
Number Systems
2
1
= 2 2
0
= 1
0 0
0 1
1 0
1 1
Equivalent Decimal
0
1
2
3
1bit
0
1
Equivalent Decimal
0
1

5. Binary Counting
3 bit & 4 bit binary counting
( Total count for 3 bit 2
3
= 8)
( Total count for 4 bit 2
4
= 16)
Number Systems
2
2
= 42
1
= 22
0
= 1
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Decimal
0
1
2
3
4
5
6
7
2
3
= 82
2
= 42
1
= 22
0
= 1
0 0 0 0
0 0 0 1
0 0 1 0
0 0 1 1
0 1 0 0
0 1 0 1
0 1 1 0
0 1 1 1
1 0 0 0
1 0 0 1
1 0 1 0
1 0 1 1
1 1 0 0
1 1 0 1
1 1 1 0
1 1 1 1
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

6. Counting of Number Systems
Octal Counting
0,1,2,3,4,5,6,7,
10,11,12,13,14,15,16,17, 8/8 = 10
20,21,22,23,24,25,26,27, 17/8 = 21
30....................
Hexa Decimal Counting
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,
10,11,12,13,14,15,16,17,18,19,1A,1B,1C,1D,1E,1F, 27/16 = 1B
20,21,22,23,24,25,26,27,28,29,2A,2B,2C,2D,2E,2F, 38/16 = 26
30...........................
Number Systems

7. ConversionofAnyNumberSystemintoDecimalNumber
Decimal Number System
Convert into Decimal
Number Systems Conversions

8. Conversion of Any Number System into Decimal Number
Cont...
Convert into Decimal
Number Systems Conversions

9. Conversion of Decimal Number into Any Radix Number
It is performed in 2 Steps
1.Conversion of Integer Part –successive division method
2.Conversion of Fractional Part –successive multiplication method
Successive Division for Integer Part conversion
Successive multiplication for Fractional Part conversion
Number Systems Conversions

10. Perform the following Conversions
Convert the decimal number 37 into its binary
equivalent
37
10
=
( 1 0 0 1 0 1)
2
Number Systems Conversions

11. Perform the following Conversions
Convert the decimal number 214 into its octal
equivalent
214
10
=
( 326)
8
Number Systems Conversions

12. Perform the following Conversions
Convert the decimal number 3509 into its
hexadecimal equivalent
3509
10
=
( DB5)
H
3509
10
=
( DB5)
16
Number Systems Conversions

13. Perform the following Conversions
Convert the decimal number 54 into radix 4
54
10
=
( 312)
4
Number Systems Conversions

14. Perform the following Conversions
Convert the decimal number .8125 into its binary
equivalent
.8125
10
=
( .1101)
2
Number Systems Conversions

15. Perform the following Conversions
Convert the decimal number .45 into its Octal
equivalent
.45
10
=
( .34631)
8
Number Systems Conversions

16. Perform the following Conversions
Convert the decimal number .64 into its Hex
equivalent
.64
10
=
( . A3D7)
16
Number Systems Conversions

17. Perform the following Conversions
Convert the decimal number 37.8125 into its binary
equivalent
37
10
=
( 1 0 0 1 0 1. 1 1 0 1)
2
Number Systems Conversions

18. Perform the following Conversions
Convert the decimal number 214.45 into its octal
equivalent
214.45
10
=
( 326.34631)
8
Number Systems Conversions

19. Perform the following Conversions
Convert the decimal number 3509.64 into its
hexadecimal equivalent
3509.64
10
=
( DB5.A3D7)
16
Number Systems Conversions

20. Binary to octal
8 = 2
3
Number Systems Conversions

21. Binary to Hexadecimal
16 = 2
4
Number Systems Conversions

22. Octal to Binary
Number Systems Conversions

23. Hexadecimal to Binary
Number Systems Conversions

24. Octal to Hexadecimal
This can be performed in 2 steps
•Octal to Binary conversion
•Binary to Hexadecimal conversion
Number Systems Conversions

25. Hexadecimal to Octal
This can be performed in 2 steps
•Hexadecimal to Binary conversion
•Binary to Octal conversion
Number Systems Conversions

26. Binary Arithmetic
•Binary Addition
•Binary Subtraction
•Binary Multiplication
•Binary Division
Note:

27. Binary Addition
Perform the binary addition
Binary Arithmetic

28. Binary Subtraction
Perform the binary subtraction
Binary Arithmetic

29. Binary Multiplication
Perform the binary multiplication
Binary Arithmetic

30. Perform the binary multiplication
Binary Arithmetic

31. Perform the binary multiplication
Binary Arithmetic

32. Binary Division
Perform the binary division
Binary Arithmetic

33. Perform the binary division
Binary Arithmetic

34. Complements
Binary :
•r complement –2’s complement
•( r-1) complement -1’s
complement
Decimal
•r complement –10’s complement
•( r-1) complement -9’s
complement
There are two types of complements for each base-rsystem:
•The radix complement –r complement
•The diminished radix complement -( r-1) complement
Octal :
•r complement –8’s complement
•( r-1) complement -7’s
complement
Hexadecimal
•r complement –16’s complement
•( r-1) complement -15’s
complement
Given a number Nin base rhaving ndigits, the (r–1)’s complement of Nis defined as:
(r
n
–1) –N
Binary Arithmetic

35. Complements Examples
Binary Arithmetic

36. Subtraction of Binary Numbers using
Complement Addition
Let the given Number is X -Y
Using 1’s complement
•Find the 1’s complement of Y
•Add with X
•If carry =1, then add the carry with the LSB of the result ( This is called end around
carry)
•Check MSB, if MSB is 0 it represent result is positive and it’s final result. If MSB is 1
it represent result is negative, so 1’s complement of result is final result.
Using 2’s complement
•Find the 2’s complement of Y
•Add with X
•If carry =1, then remove the carry ( This is called discard carry)
•Check MSB, if MSB is 0 it represent result is positive and it’s final result. If MSB is 1
it represent result is negative, so 2’s complement of result is final result.
Binary Arithmetic

37. Given two numbers A= 1010100 and B=1000011, Perform
subtraction (a) A-B (b) B-A using 1’s complement method
No.of digits in A and B must be equal
Binary Arithmetic

38. Given two numbers A= 1010100 and B=1000011, Perform
subtraction (a) A-B (b) B-A using 2’s complement method
Binary Arithmetic

39. Home work
1.Given two numbers X= 111010 and Y=10011,
Perform subtraction (a) X-Y (b) Y-X using 1’s and 2’s
complement method
2. Given two numbers X= 11010.01 and Y=10001.11,
Perform subtraction (a) X-Y (b) Y-X using 1’s and 2’s
complement method
Note :No.of digits in A and B must be equal
Binary Arithmetic

40. Binary Codes

41. Binary Codes

42. Gray code
Only one bit
change in two
consecutive
numbers
Binary Codes

43. Binary to Gray code conversion
Binary Codes

44. Gray to Binary code conversion
Binary Codes

45. Convert the following binary numbers into
Gray
Binary Codes

46. Convert the following Gray numbers into
binary
Binary Codes

47. Logic Functions

48. Basic Logic Gates
Logic Functions

49. Basic Logic Gates
Logic Functions

50. Universal Gates
1. NAND Gate-
A NAND Gate is constructed by connecting a NOT Gate at the output terminal of the
AND Gate.
The output of NAND gate is high (‘1’) if at least one of its inputs is low (‘0’).
The output of NAND gate is low (‘0’) if all of its inputs are high (‘1’).
A B Y = (A.B)’
0 0 1
01 1
1 0 1
1 1 0
Truth Table
Logic Symbol-
Logic Functions

51. Universal Gates
2. NOR Gate-
A NOR Gate is constructed by connecting a NOT Gate at the output terminal of the
OR Gate.
The output of OR gate is high (‘1’) if all of its inputs are low (‘0’).
The output of OR gate is low (‘0’) if any of its inputs is high (‘1’).
Truth Table
A B Y = A + B
0 0 1
0 1 0
1 0 0
1 1 0
Logic Symbol-
Logic Functions

52. Boolean Operators
Boolean Algebra
Boolean function relates the input and output variable
with the operators

53. Boolean Algebra

54. Properties of Boolean Algebra
Boolean Algebra

55. Consensus Theorem
Boolean Algebra

56. Problems
Simplify the following Boolean expression
Boolean Algebra

57. Simplify the following Boolean expression
Boolean Algebra

58. Simplify the following Boolean expression
Boolean Algebra

59. Simplify the following Boolean expression
Boolean Algebra

60. Simplify the following Boolean expression
Boolean Algebra

61. Home Work
Simplify the following Boolean expression
Boolean Algebra

62. NOT function using NAND gate
Functionally complete operation sets

63. AND function using NAND gate
Functionally complete operation sets

64. OR function using NAND gate
Functionally complete operation sets

65. NOR function using NAND gate
Complement of OR is NOR
Functionally complete operation sets

66. NOT function using NOR gate
Functionally complete operation sets

67. OR function using NOR gate
Functionally complete operation sets

68. AND function using NOR gate
Functionally complete operation sets

69. NAND function using NOR gate
Complement of AND is NAND
Functionally complete operation sets

70. Realization of switching function

71. Realization of switching function

72. Realisation of switching function using
NAND / NOR gates

74. Cont...