Title: Combinational Logic Circuits
1CombinationalLogic Circuits
2CombinationalLogic Circuits
- Binary Logic and Gates
- Boolean Algebra
- Standard Forms
- Map Simplification
- NAND and NOR Gates
- Exclusive-OR Gates
- Integrated Circuits
3Digital Logic Gates
4Gates with More than Two Inputs
5CombinationalLogic Circuits
- Binary Logic and Gates
- Boolean Algebra
- Standard Forms
- Map Simplification
- NAND and NOR Gates
- Exclusive-OR Gates
- Integrated Circuits
6Basic Identities of Boolean Algebra
7Implementation of Boolean Function with Gates
8CombinationalLogic Circuits
- Binary Logic and Gates
- Boolean Algebra
- Standard Forms
- Map Simplification
- NAND and NOR Gates
- Exclusive-OR Gates
- Integrated Circuits
9Minterms for Three Variables
10Sum 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
11Sum 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)
12Sum of Products Exclusive-OR
!X Y
Z (!X Y) (X !Y)
X !Y
13Maxterms for Three Variables
14Product of Sums Design
Maxterms A maxterm is NOT a minterm maxterm M0
NOT minterm m0 M0 m0 (X . Y) (X
Y) X Y
15Product 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
16Product 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
17Product 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)
18Product of Sums Exclusive-OR
19Three- Level and Two- Level Implementation
20CombinationalLogic Circuits
- Binary Logic and Gates
- Boolean Algebra
- Standard Forms
- Map Simplification
- NAND and NOR Gates
- Exclusive-OR Gates
- Integrated Circuits
21Two-Variable Map
22Three-Variable Map
23Three- Variable Map Flat and on a Cylinder to
Show Adjacent Squares
24Three-variable K-Maps
1
1
1
1
F !X !Y X Z
25Three-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
26Three-variable K-Maps
1
1
1
1
1
F Y !Z X
27Three-variable K-Maps
1
1
1
1
1
1
F !X !Y X y Z
28Three-variable K-Maps
1
1
1
1
F X Z !X !Z
29Three-variable K-Maps
1
1
1
1
1
1
F Y !Z
30Three-variable K-Maps
1
1
1
1
F m0 m2 m5 m7 S(0,2,5,7)
31Four-Variable Map
32Four-Variable Map Flat and on a Torus to Show
Adjacencies
33Four-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
34Four-variable K-Maps
F(W,X,Y,Z) S(2,4,5,6,7,9,13,14,15)
35Four-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
36CombinationalLogic Circuits
- Binary Logic and Gates
- Boolean Algebra
- Standard Forms
- Map Simplification
- NAND and NOR Gates
- Exclusive-OR Gates
- Integrated Circuits
37Prime 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.
38CombinationalLogic Circuits
- Binary Logic and Gates
- Boolean Algebra
- Standard Forms
- Map Simplification
- NAND and NOR Gates
- Exclusive-OR Gates
- Integrated Circuits
39Digital Logic Gates
gt
40gt
41Logical Operations with NAND Gates
42Alternative Graphics Symbols for NAND and NOT
Gates
43Logical Operations with NOR Gates
44Two Graphic Symbols for NOR Gate
45Generalized 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))
46F !(!(X Y) !(X Z) !(Y Z))
47F !(!(X Y) !(X Z) !(Y Z))
NAND Gate
48F X Y X Z Y Z
X
Y
X
F
Z
Y
Z
49CombinationalLogic Circuits
- Binary Logic and Gates
- Boolean Algebra
- Standard Forms
- Map Simplification
- NAND and NOR Gates
- Exclusive-OR Gates
- Integrated Circuits
50Exclusive-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
51Exclusive-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
52Parity Generation and Checking
53CombinationalLogic Circuits
- Binary Logic and Gates
- Boolean Algebra
- Standard Forms
- Map Simplification
- NAND and NOR Gates
- Exclusive-OR Gates
- Integrated Circuits
54Fully 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.