1. ELL100: INTRODUCTION TO ELECTRICAL ENGG.
Course Instructors:
J.-B. Seo, S. Srirangarajan, S.-D. Roy, and S. Janardhanan
Department of Electrical Engineering, IITD

2. Number systems Decimal, Octal, Hex, binary systems
Decimal number system
Octal (0,1,…,7): Base 8
= 448
= 40
= 4

3. Number systems Decimal, Octal, Hex, binary systems
Hex (0,1,…9,A, B, C,…,F): Base 16
Binary
= 256
= 224

4. Most significant & least significant bit
For any number system (non-fractional part)
Right most: Least significant digit/bit (LSB)
Left most: Most significant digit/bit (MSB)
Decimal:
Binary:
Hex: AE143

5. Conversion from Decimal to Base N
Divide the decimal number by N. Reminder is LSB
If the quotient is not divisible, the conversion is done.
Else, repeat step 1 using the quotient as a new decimal number.
New reminder is the next LSB 30 1 27 13 6 3 1 16
492 16 30 2 55 2 27 2 13 2 6 2 3
480 16 54 26 12 6 2 12
···· C 14
···· E 1 1 1 1

6. Ones’ complement (1’s complement)
Suppose a 8-bit (binary) integer
How can we express – 127?
To obtain the 1’s complement of a binary number: flip all the bits
: the MSB denotes the negative sign

7. One’s complement value
Bits
Unsigned value
One’s complement value 1 2
126
127
128
–127
129
–126
253 –2
254 –1
255 –0

8. Ones’ complement
Adding two values:
Align the values on the LSB
Add, propagating any carry to the bit one position left.
If the carry extends past the end of the word, it is said to have "wrapped around", a condition called an “end-around carry”
When this occurs, the bit must be added back in at the right-most bit. 22 22 + 3 +
– 0 25 1 1 22

9. Ones’ complement 6 – 19 = 19 + 3 = 19 – (-3) = 0000 0110 6 0001 0011
Subtraction uses “end-around borrow” if necessary. Subtract it from the LSB
6 – 19 =
= 19 – (-3) = 6 19 – 19 –
– 3 1 25 1 1 1
–13 22

10. Two’s complement (2’s complement)
Two’s complement of an N-bit number:
Its complement with respect to 2N
 The two’s complement of three-bit number 010
1000
= 1000 –
010
110
 Find one’s complement and add `1’ to the LSB
= 110
– 2
 No negative zero:
000
111
+ 001
= 000

11. Two’s complement
Decimal value
Binary (2’s compl.)
Two’s complement 1 2
126
127
–128
–127
–126 –2 –1

12. Two’s complement (2’s complement)
Adding two’s complement numbers requires no special processing
15 – 5 = (-5)
15 – (-5) =
borrow 15 15 +
– 5 –
– 5 10 20
Carry bit is ignored

13. Two’s complement (2’s complement)
Overflow =
(-103) + (-69) = -172
104
– 103 + 45 –
– 69
149
If the sum of two positive numbers yields a negative result, the sum has overflowed.
If the sum of two negative numbers yields a positive result, the sum has overflowed.
Otherwise, the sum has not overflowed.

14. Gates Truth table Truth table
Simplest Logic Gate – also known as the complement or the invertor
Truth table
Simplest Logic Gate – also known as the complement or the invertor
Truth table

15. Gates
AND gate
OR gate
Truth table
Truth table

16. Gates
NAND gate
NOR gate
Truth table
Truth table

17. Gates
XOR Gate
XNOR
Truth table
Truth table

18. Boolean Algebra – 1

19. Boolean Algebra – 2

20. Boolean Algebra – 2

21. Boolean Algebra – 2

22. DeMorgan’s theorem
Example =

23. Logic circuit analysis

24. Karnaugh map (K-map) The sum of products (minterm)
Three-variable Karnaugh map

25. Karnaugh map (K-map) The sum of products (minterm)
Three-variable Karnaugh map

26. Karnaugh map (K-map) The sum of products (minterm)
Three-variable Karnaugh map

27. Karnaugh map (K-map) The sum of products (minterm)
Three-variable Karnaugh map

28. Karnaugh map (K-map)

29. Karnaugh map (K-map)

30. Karnaugh map (K-map)

31. Karnaugh map (K-map)

32. Karnaugh map (K-map)
K-map with four variables

33. Example – 1
Design a circuit to take a 4-bit number ABCD and produce a single output Y that is true only if the input represents a prime number

34. Example – 1
Design a circuit to take a 4-bit number ABCD and produce a single output Y that is true only if the input represents a prime number or 0, and 1.

35. Example – 1
Design a circuit to take a 4-bit number ABCD and produce a single output Y that is true only if the input represents a prime number

36. Example – 1
Design a circuit to take a 4-bit number ABCD and produce a single output Y that is true only if the input represents a prime number

37. Example – 2
LED seven-segment display is used to indicate decimal digits, 0,1,…9. a f b g e c 8 9 d 6 7 4 5 2 3 1

38. Example – 2
LED seven-segment display is used to indicate decimal digits, 0,1,…9. a f b g e c 8 9 d 6 7 4 5 2 3 1

39. Example – 2
LED seven-segment display is used to indicate decimal digits, 0,1,…9. a f b g e c 8 9 d 6 7 4 5 2 3 1

40. Example – 2 1 1 1 1 1 1 x x x x x x 1 1
Don’t care condition

41. Example – 2 1 1 1 1 1 1 x x x x x x 1 1

42. Example – 2

43. Karnaugh map (K-map) The sum of products (minterm)
The product of sums (maxterm)
(_ _ _) + (_ _) + (_ _ _ )
(_ + _ + _)(_ + _)(_ + _ + _ )
Minterm
Maxterm

44. Karnaugh map (K-map)

45. Karnaugh map (K-map) To convert a Boolean function F from SoP to PoS
Find in SoP form
Find in PoS form

46. Karnaugh map (K-map)

47. Karnaugh map (K-map)

48. Karnaugh map (K-map)

49. Karnaugh map (K-map)

50. Karnaugh map (K-map)

51. Karnaugh map (K-map)

52. Karnaugh map (K-map)

53. Karnaugh map (K-map)

54. Karnaugh map (K-map)

55. Karnaugh map (K-map)

56. Half-Adder

57. Full-Adder

58. Full-Adder

59. Full-Adder
 Sum
 Carry-out

60. Full-Adder

61. 4-bit Binary Adder How can you build ? A3A2A1A0 + B3B2B1B0
S3S2S1S0
Cout, Cin
How can you build ?
Four 1-bit FA Or
Eight 1-bit HA

62. 4-bit Binary Adder How can you build ? A3A2A1A0 + B3B2B1B0
S3S2S1S0
Cout, Cin
How can you build ?
Four 1-bit FA Or
Eight 1-bit HA

63. 4-bit Binary Adder How can you build ? A3A2A1A0 + B3B2B1B0
S3S2S1S0
Cout, Cin
How can you build ?
Four 1-bit FA Or
Eight 1-bit HA

64. 4-bit Binary Adder How can you build ? A3A2A1A0 + B3B2B1B0
S3S2S1S0
Cout, Cin
How can you build ?
Four 1-bit FA Or
Eight 1-bit HA

65. 4-bit Binary Adder How can you build ? A3A2A1A0 + B3B2B1B0
S3S2S1S0
Cout, Cin
How can you build ?
Four 1-bit FA Or
Eight 1-bit HA

66. 4-bit Binary Adder How can you build ? A3A2A1A0 + B3B2B1B0
S3S2S1S0
Cout, Cin
How can you build ?
Four 1-bit FA Or
Eight 1-bit HA

67. 4-bit Binary Adder Four 1-bit FA Or Eight 1-bit HA How can you build ?
A3A2A1A0
+ B3B2B1B0
S3S2S1S0
Cout, Cin
How can you build ?
Four 1-bit FA Or
Eight 1-bit HA

68. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum

69. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum

70. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum

71. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum

72. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum

73. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum

74. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum
Select bit (M)

75. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum
Select bit (M) = 1  ?

76. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum
Select bit (M) = 1  output flips B

77. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum
Select bit (M) = 1  output flips B
Select bit (M) = 0  output = B

78. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum
Select bit (M) = 1  Enable Subtract
Select bit (M) = 0  Enable Add

79. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum
Select bit (M) = 1  Enable Subtract
Select bit (M) = 0  Enable Add

80. Design a 4-bit adder that can also subtract the two numbers ?
A3A2A1A0
-/+ B3B2B1B0
Design a 4-bit adder that can also subtract the two numbers ?
Algorithm
You will need a selection bit to select between the 2 operations – ADD or SUB
If SUB, then first Flip all the bits of B, and find B’
Add B’ to A (like the normal ADDER)
Add 1 to the Sum
Select bit (M) = 1  Enable Subtract
Select bit (M) = 0  Enable Add