acoe1611 combinational logic circuits reference: m. mano, c. kime, “logic and computer design...
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ACOE161 1
Combinational Logic Circuits
Reference: M. Mano, C. Kime, “Logic and Computer Design Fundamentals”, Chapter 2
ACOE161 2
Basic Logic Gates
Buffer AND OR EX-OR
A X X X XA A A
BBB
X = A X = A B X = A + B X = A + B
A X0 0
11
A X0
0
1
10
001
1 1
0
0
B A X0
1
1
10
001
1 1
0
1
B A X0
1
0
10
001
1 1
0
1
B
Truth Table
Logic Expression
Gate Symbol
Logic Function
ACOE161 3
Basic Logic Gates with Inverted Outputs
NOT NAND NOR EX-NOR
A X X X XA A A
BBB
X = A X = A B X = A + B X = A + B
A X0
01
1
A X0
01
10
001
1 1 00
B A X0 1
1 10
001
1 1
01
B A X0 1
010
001
1 1
0
1
B
ACOE161 4
Logic Gates with more than two inputs
3-Input AND Gate
3-Input OR Gate
4-Input AND Gate 4-Input OR Gate
ACOE161 5
Circuit Implementation of a Logic Expression with Gates
X = A + BC
Logic Function A
B
C BC
BX
Logic Diagram with Gates
X = (A + B)C
Logic Function A
B
C
A+BB
X
Logic Diagram with Gates
ACOE161 6
Circuit Implementation of Logic Expressions:- Examples
X = A(B+C)+BC
Logic Function
Logic Diagram with GatesExample 1
X = (AB+C)B+C
Logic Function
Logic Diagram with GatesExample 2
ACOE161 7
Circuit Implementation of Logic Expressions:- Homework
X = (AB+C)(B+C)
Logic Function
Logic Diagram with GatesHomework 1
X = (ABC+C)B+AC
Logic Function
Logic Diagram with GatesHomework 2
ACOE161 8
Truth Tables
Truth table of a logic circuit is a table showing all the possible input combinations with the
corresponding value of the output. Examples:
A B C
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
X
0
0
0
1
1
1
1
0
(a) Show the truth table of a 3-input circuitthat gives at its output a logic 1 if the inputforms a number between 3 and 6.
Inputs Output
0
1
2
3
4
5
6
7
(b) Show the truth table of the logic expression:
A B C
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
AB C AB + C
0
0
0
0
1 1
0
1
1
0
1
1
0
1
0 0
0
00
1 1
1 0 1
X = (AB+C)(A+C)
1
1
1
1
0
0
0
1
1
1
1
1
0
0
1
1
A + C
X = (AB + C)(A + C)
ACOE161 9
Truth Tables: Examples
X = (AB + C)(A + C)
A B C
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
A B C
1
1
1
1
1 1
1
1
1
1
1
1
0
0
0 0
0
00
0 0
0 0 0
ACOE161 10
Minterms and maxterms
ACOE161 11
Standard forms: Sum of Products
X = ABC + ABC + ABC + ABC
A B C
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
A B C
1
1
1
1
1 1
1
1
1
1
1
1
0
0
0 0
0
00
0 0
0 0 0
ACOE161 12
Logic expression and truth table of a logic circuit
AB
CX
T1 =
T2 =
T3 =
T4 =
A B C
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
1
2
3
4
5
6
7
T1 T2 T3 T4 X
Logic Expression: X =
SoP Form: X =
PoS Form: X =
ACOE161 13
Example: Find the logic expression and fill up the truth table for the circuit below.
AB
CX
T1 =
T2 =
T3 =
T4 =
A B C
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
1
2
3
4
5
6
7
T1 T2 T3 T4 X
X =
ACOE161 14
Homework: Find the logic expression and fill up the truth table for the circuit below.
AB
C
X
T1=
T2 =
T3 =
T4 =
A B C
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
1
2
3
4
5
6
7
T1 T2 T3 T4 X
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
8
9
10
11
12
13
14
15
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0T5 =D
T5D
Logic Expression: X =
SoP Form: X =
PoS Form: X =
ACOE161 15
Analyzing a logic circuit using timing diagrams
A
B
CX
T1 =T2 =
T3 =
T4 =
A B C
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
1
2
3
4
5
6
7
T1 T2 T3 T4 X
A
B
C
X
Logic 0
Logic 1
ACOE161 16
Homework: Fill up the truth table and timing diagram for the circuit below.
A
B
C
X
T1 =T2 =
T3 =
T4 =A B C
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
1
2
3
4
5
6
7
T1 T2 T3 T4 X
A
B
C
X
T5 =
T5
ACOE161 17
Boolean Algebra
1. X 0 X 3. X 1 1
5. X X X
7. X X 1
2. X 0 0
4. X 1 X
6. X X X
8. X X 0
9. X X 10. X X11. X( XZY Z XY ) 12. X + Z)YZ X Y X ( )(
13. X +Y X Y 14. X Y X Y15. X Y + XY XY 16. X Y + X Y XY
17. X + XY +YX 18. X + XY +YX
Basic Boolean identities:
ACOE161 18
Boolean Algebra (Examples)
1. X XY X
3. XY XY X
5. XY XZ + YZ = XY + XZ
7. AB +AB+AB 1
2. X(X + Y) X
4. (X +Y)(X +Y) X
6. (X + Y)(X X + Y)(X Z Y Z Z)( ) ( )
8. XYZ X +Y + Z
9. AB AB AB A B 10. AB A B AB AB
Prove the following identities using Boolean algebra and truth tables:
ACOE161 19
Digital circuit simplification using Boolean algebra
• Logic functions are simplified in order to reduce the number of gates required to implement them. Thus the circuit will
– cost less,– need less space and power, – be build faster with less effort.
• For example the expression F needs six gates to be build. If the expression is simplified then the function can be implemented with only two gates.
F = XYZ + XYZ XYZ YZ(X X) XYZ YZ XYZ Y(Z XZ) F Y(Z X)
F = XYZ+ XYZ XYZ F Y(Z X)
X
Y
Z
F
F
XZ
Y
ACOE161 20
Boolean Algebra (Examples)
1. F = XY Z XYZ XZ 2. F = X YZ + XYZ + Y
Simplify the expressions given below. Use truth tables to verify your results.
X Y Z
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
X Y Z
1
1
1
1
1 1
1
1
1
1
1
1
0
0
0 0
0
00
0 0
0 0 0
X Y Z
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
X Y Z
1
1
1
1
1 1
1
1
1
1
1
1
0
0
0 0
0
00
0 0
0 0 0
ACOE161 21
Boolean Algebra (Examples - Cont.)
3. F = (X +Y + Z X Y Z)( ) 4. F = (X +Y + Z)(X + Z)
Simplify the expressions given below. Use truth tables to verify your results.
X Y Z
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
X Y Z
1
1
1
1
1 1
1
1
1
1
1
1
0
0
0 0
0
00
0 0
0 0 0
X Y Z
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
X Y Z
1
1
1
1
1 1
1
1
1
1
1
1
0
0
0 0
0
00
0 0
0 0 0
ACOE161 22
Boolean Algebra (Examples - Cont.)
5. F = XY X YZ + (Y + Z)
Simplify the expression given below. Use truth tables to verify your results.
X Y Z
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
X Y Z
1
1
1
1
1 1
1
1
1
1
1
1
0
0
0 0
0
00
0 0
0 0 0
ACOE161 23
Boolean Algebra (Examples - Cont.)
6. F = (X + Y X)( )( ) Z Y Z
Simplify the expression given below. Use truth tables to verify your results.
X Y Z
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
X Y Z
1
1
1
1
1 1
1
1
1
1
1
1
0
0
0 0
0
00
0 0
0 0 0
ACOE161 24
Digital circuit Examples with 1 output