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A lecture on the implementation of digital circuits using nand and nor gates instead of and and or gates. It covers the logic operations, graphic symbols, and procedures for implementing boolean functions using two and multilevel nand and nor gates. The document also discusses the differences between nand and nor gates and their applications.
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Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009
Graphic Symbols
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Page 2
NAND Implementation
Inverter
Logic Operations with NAND gates
Bubble denotes complementation
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Two^ graphic^ symbols^ for^ the^ NAND^ gates Page 3
AND‐ Invert Invert‐ OR
Two Level Implementation
The implementation of Boolean functions with NAND gates requires that the functions be in sum of products (SOP) form.
The Rule
This function can be implemented by three different ways as shown in the circuit diagram a, b, c
The function is implemented in ( )
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Page 4
The function is implemented in (a) With AND and OR gates (b) The AND gates are replaced by NAND gates and the OR gate is replaced by a NAND gate with invert‐OR graphic. (c) It is similar to case b) but the output NAND gate is redrawn with the AND‐invert.
F = ( AB CD. )
Example 1: Consider the Boolean function, implement the circuit diagram by using multilevel
NAND gate.
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Page 7
Example 2: Consider the following Boolean function, implement the circuit diagram by using
AND‐OR gates
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Page 8
NAND gates
•The NOR operation is the dual of the NAND operation •The NOR gate is another universal gate to implement any Boolean function •Easy for OR‐AND (product of sums)
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Page 9
Logic Operations with NOR gates
Two graphic symbols for the NOR gates
OR‐ Invert Invert‐AND
Example1: Consider the following Boolean function, implement the circuit diagram by using two l;evel NOR gate
The implementation of Boolean functions with NOR gates requires that the functions be in product of sums (POS) form.
The Rule
l;evel NOR gate.
Transformation from OR‐AND diagram to NOR diagram •OR gates => OR‐invert •AND gate => invert‐AND
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Page 10
OR‐AND gate NOR^ gate
AND‐OR‐INVERT Implementation
AND‐NOR = NAND‐AND = AND‐OR‐INVERT
F = (AB + CD + E)’
SiSimilar il to AND OR AND OR INVERTAND‐OR, AND‐OR‐INVERT requiresi an expressioni ii n sum off productsd Given F, we can implement F’ with AND‐OR‐INVERT
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Page 13
OR‐AND‐INVERT Implementation
OR‐NAND = NOR‐OR = OR‐AND‐INVERT F = [ (A+B) (C+D) E ] ’
Similar to OR‐AND, OR‐AND‐INVERT requires an expression in products of sum GiGiven FF, we can ii mplementl F’F’ withi h OR AND INVERT OR‐AND‐INVERT
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Page 14
Example: Implement the function in Fig.1 with the four two level
form.
F = (F’)’ = (x’y + xy’ + z)’
Fig.
= (x + y’)(x’ + y)z F’ = (F)’ = (x’y’z’ + xyz’)’ = (x + y + z)(x’ + y’ + z)
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Page 15
Exclusive‐OR (XOR) Function
(x ْ y)’ = (xy’ + x’y)’ = (x’ + y)(x + y’) = x’y’ + xy(x y)(x y ) x y xy
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Page 16
Communtative: A ْ B = B ْ A Associative: (A ْ B) ْ C = A ْ (B ْ C) = A ْ B ْ C Only a limited number of Boolean functions can be expressed in terms of XOR operations, but it is particularly useful in arithmetic operations and error‐detection and correction circuits
Map for a 4 ‐variable XOR Function
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Page 19
Exercises
b)
A B + ABD + AB D + AC D + ABC
F A B C D ( , , , ) = (^) ∑(0,1, 2,3, 4,8,9,12)
( A B + C D ) E + BC ( A + B )
Dr. Saied M. Abd El‐atty Soliman ssoliman1@ksu.edu.sa
14 April 2009 Page 20
F W ( , X Y , , Z ) = (^) ∑(0, 4,8,9,10,11,12,14)
( ) ( )
