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CombinationalLogic Circuits
Chapter 2
Mano and Kime
CombinationalLogic Circuits
• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits
Digital Logic Gates
*
Gates with More than Two Inputs
CombinationalLogic Circuits
• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits
Basic Identities of Boolean Algebra
Implementation of Boolean Function with Gates
CombinationalLogic Circuits
• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits
Minterms for Three Variables
Sum of Products Design
X Y minterms0 0 m0 = !X & !Y0 1 m1 = !X & Y1 0 m2 = X & !Y1 1 m3 = X & Y
Sum of Products Design
X Y Z0 0 00 1 11 0 11 1 0
Design an XOR gate
m1 = !X & Ym2 = X & !Y
Z = m1 + m2 = (!X & Y) + (X & !Y)
X Y
X !X Y !Y
Sum of Products: Exclusive-OR
!X & Y
X & !Y
Z = (!X & Y) + (X & !Y)
Maxterms for Three Variables
Product of Sums Design
Maxterms:A maxterm is NOT a mintermmaxterm M0 = NOT minterm m0M0 = m0’ =(X’ . Y’)’ = (X + Y)” = X + Y
Product of Sums Design
X Y minterms maxterms0 0 m0 = !X . !Y M0 = !m0 = X + Y0 1 m1 = !X . Y M1 = !m1 = X + !Y1 0 m2 = X . !Y M2 = !m2 = !X + Y1 1 m3 = X . Y M3 = !m3 = !X + !Y
Product of Sums Design
X Y Z0 0 00 1 11 0 11 1 0
Design an XOR gate
Z is NOT minterm m0
AND it is NOT minterm m3
Product of Sums Design
X Y Z0 0 00 1 11 0 11 1 0
Design an XOR gate
M0 = X + Y
M3 = !X + !Y
Z = M0 & M3 = (X + Y) & (!X + !Y)
X Y
X !X Y !Y
X + Y
!X + !Y
Z
Z = (X + Y) & (!X + !Y)
Product of Sums: Exclusive-OR
Three- Level and Two- Level Implementation
CombinationalLogic Circuits
• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits
Two-Variable Map
Three-Variable Map
Three- Variable Map: Flat and on a Cylinder to Show Adjacent Squares
Three-variable K-Maps
XYZ
00 01 11 10
0
1
1 1
1 1
F = !X & !Y + X & Z
Three-variable K-Maps
XYZ
00 01 11 10
0
1
1 1
1 1
F = !X & !Y & !Z + !X & !Y & Z + X & !Y & Z + X & Y & Z
F = !X & !Y & (!Z + Z) + X & Z & (!Y + Y) = !X & !Y + X & Z
Three-variable K-Maps
XYZ
00 01 11 10
0
1 1
1
1 1
F = Y & !Z + X
1
Three-variable K-Maps
XYZ
00 01 11 10
0
1
1 1
1 11
1
F = !X & !Y + X & y + Z
Three-variable K-Maps
XYZ
00 01 11 10
0
1
1 1
1 1
F = X & Z + !X & !Z
Three-variable K-Maps
XYZ
00 01 11 10
0
1
1 1
1 1
1
1
F = Y + !Z
Three-variable K-Maps
X
YZ00 01 11 10
0
1
0 1 23
4 5 67
1 1
1 1
F = m0 + m2 + m5 + m7 = (0,2,5,7)
Four-Variable Map
Four-Variable Map: Flat and on a Torus to Show Adjacencies
Four-variable K-Maps
WXYZ
00 01 11 10
00
01
11
10
0 1 3 2
4 5 7 6
8 9
13 15 14
11 10
12
Each square is numbered in the above K-map
Four-variable K-Maps
WXYZ
00 01 11 10
00
01
11
10
0 1 23
4 5 67
8 9 1011
12 13 1415
F(W,X,Y,Z) = (2,4,5,6,7,9,13,14,15)
Four-variable K-Maps
1 11
1
1
WXYZ
00 01 11 10
00
01
11
10
111
1
F = !W & X + X & Y + !W & Y & !Z + W & !Y & Z
CombinationalLogic Circuits
• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits
Prime Implicants
F = XY’Z + X’Z’ + X’Y
Each product termis an implicant
A product term that cannot have any of itsvariables removed and still imply the logicfunction is called a prime implicant.
CombinationalLogic Circuits
• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits
Digital Logic Gates
>
>
��Logical Operations with NAND Gates
Alternative Graphics Symbols for NAND and NOT Gates
��Logical Operations with NOR Gates
Two Graphic Symbols for NOR Gate
Generalized De Morgan’s 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))
X
Y
X
Z
Y
Z
F
F = !(!(X & Y) & !(X & Z) & !(Y & Z))
X
Y
X
Z
Y
Z
F
F = !(!(X & Y) & !(X & Z) & !(Y & Z))
NAND Gate
X Y
X Z
Y Z
F
F = X & Y + X & Z + Y & Z
CombinationalLogic Circuits
• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits
Exclusive-OR Gate
XOR
X
YZ
Z = X $ Y
X Y Z0 0 00 1 11 0 11 1 0
X $ 0 = XX $ 1 = !XX $ X = 0X $ !X = 1
X $ !Y = !(X $ Y)!X $ Y = !(X $ Y)A $ B = B $ A(A $ B) $ C = A $ (B $ C)
= A $ B $ C
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
Parity Generation and Checking
CombinationalLogic Circuits
• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
• Map Simplification
• NAND and NOR Gates
• Exclusive-OR Gates
• Integrated Circuits
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.