Combinational Logic Circuits

1
CombinationalLogic Circuits

• Chapter 2
• Mano and Kime

2
CombinationalLogic Circuits

• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits

3
Digital Logic Gates

4
Gates with More than Two Inputs
5
CombinationalLogic Circuits

• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits

6
Basic Identities of Boolean Algebra
7
Implementation of Boolean Function with Gates
8
CombinationalLogic Circuits

• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits

9
Minterms for Three Variables
10
Sum of Products Design
X Y minterms 0 0 m0 !X !Y 0 1 m1
!X Y 1 0 m2 X !Y 1 1 m3 X Y
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Sum of Products Design
Design an XOR gate
X Y Z 0 0 0 0 1 1 1 0 1 1 1 0
m1 !X Y m2 X !Y
Z m1 m2 (!X Y) (X !Y)
12
Sum of Products Exclusive-OR
!X Y
Z (!X Y) (X !Y)
X !Y
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Maxterms for Three Variables
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Product of Sums Design
Maxterms A maxterm is NOT a minterm maxterm M0
NOT minterm m0 M0 m0 (X . Y) (X
Y) X Y
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Product of Sums Design
X Y minterms maxterms 0 0 m0 !X
. !Y M0 !m0 X Y 0 1 m1 !X . Y M1
!m1 X !Y 1 0 m2 X . !Y M2 !m2
!X Y 1 1 m3 X . Y M3 !m3 !X !Y
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Product of Sums Design
Design an XOR gate
X Y Z 0 0 0 0 1 1 1 0 1 1 1 0
Z is NOT minterm m0 AND it is NOT minterm m3
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Product of Sums Design
Design an XOR gate
X Y Z 0 0 0 0 1 1 1 0 1 1 1 0
M0 X Y M3 !X !Y
Z M0 M3 (X Y) (!X !Y)
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Product of Sums Exclusive-OR
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Three- Level and Two- Level Implementation
20
CombinationalLogic Circuits

• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits

21
Two-Variable Map
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Three-Variable Map
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Three- Variable Map Flat and on a Cylinder to
Show Adjacent Squares
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Three-variable K-Maps
1
1
1
1
F !X !Y X Z
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Three-variable K-Maps
F !X !Y !Z !X !Y Z X !Y Z
X Y Z
1
1
1
1
F !X !Y (!Z Z) X Z (!Y Y)
!X !Y X Z
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Three-variable K-Maps
1
1
1
1
1
F Y !Z X
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Three-variable K-Maps
1
1
1
1
1
1
F !X !Y X y Z
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Three-variable K-Maps
1
1
1
1
F X Z !X !Z
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Three-variable K-Maps
1
1
1
1
1
1
F Y !Z
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Three-variable K-Maps
1
1
1
1
F m0 m2 m5 m7 S(0,2,5,7)
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Four-Variable Map
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Four-Variable Map Flat and on a Torus to Show
Adjacencies
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Four-variable K-Maps
1
0
2
3
6
7
4
5
15
12
13
14
11
10
9
8
Each square is numbered in the above K-map
34
Four-variable K-Maps
F(W,X,Y,Z) S(2,4,5,6,7,9,13,14,15)
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Four-variable K-Maps
YZ
00
01
11
10
WX
00
1
F !W X X Y !W Y !Z W
!Y Z
01
1
1
1
1
11
1
1
1
10
1
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CombinationalLogic Circuits

• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits

37
Prime Implicants
Each product term is an implicant
F XYZ XZ XY
A product term that cannot have any of
its variables removed and still imply the
logic function is called a prime implicant.
38
CombinationalLogic Circuits

• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits

39
Digital Logic Gates
gt
40
gt
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Logical Operations with NAND Gates
42
Alternative Graphics Symbols for NAND and NOT
Gates
43
Logical Operations with NOR Gates
44
Two Graphic Symbols for NOR Gate
45
Generalized De Morgans Theorem

• NOT all variables
• Change to and to
• NOT the result
• --------------------------------------------
• F X Y X Z Y Z
• F !((!X !Y) (!X !Z) (!Y !Z))
• F !(!(X Y) !(X Z) !(Y Z))

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F !(!(X Y) !(X Z) !(Y Z))
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F !(!(X Y) !(X Z) !(Y Z))
NAND Gate
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F X Y X Z Y Z
X
Y
X
F
Z
Y
Z
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CombinationalLogic Circuits

• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits

50
Exclusive-OR Gate
XOR
X Y Z 0 0 0 0 1 1 1 0 1 1 1 0
X
Z
Y
Z X Y
X !Y !(X Y) !X Y !(X Y) A B B
A (A B) C A (B C) A B C
X 0 X X 1 !X X X 0 X !X 1
51
Exclusive-OR Constructed with NAND gates
X (!X !Y) Y (!X !Y) X !X X !Y
Y !X Y !Y X !Y Y !X X !Y !X
Y X Y
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Parity Generation and Checking
53
CombinationalLogic Circuits

• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits

54
Fully Complementary CMOS Gate Structure and
Examples
An Integrated circuit (IC) is a silicon
semiconductor crystal, containing the components
for the digital gates. The various gates are
connected on the chip to form the IC.