Implementations of Logic Functions using Gates

Implementations of Logic Functions using Gates

The Boolean algebra is used to express the output of any combinational network. Such a network can be implemented using logic gates. Let us see the implementation of SOP and POS Boolean expressions.

1. Implementation of SOP Boolean Expression

Consider the Boolean expression

  F = AB + C+ C   

In this expression, we have three product terms with 2 literals in each product term. Thus, we can implement these product terms by using three 2–input AND gates, as shown in D Fig. 2.7.1. Expression tells that these product terms should be ORed to get the output F. We have three product terms so we have to use 3–input OR gate to obtain the sum of products. It is important to note that literals are complemented using NOT gates.

2. Implementation of POS Boolean Expression

Consider the Boolean expression

F = (A+B) (+C) (+D+E)

In this expression, we have three sum terms with 2 literals in two terms and 3 literals in one term. We can implement these sum terms by using two 2–input OR gates and one 3–input OR gate, as shown in Fig. 2.7.2. Expression tells that these product terms should be ANDed to get the output F. We have three sum terms so we have to use 3–input AND gate to obtain the product of sums. It is important to note that literals are complemented using NOT gates.

In the previous examples we have seen that simplified SOP Boolean expression can be implemented using AND–OR gates. The AND–OR implementation is a two level implementation. In the first level we implement all product terms using AND gates and in the second level all product terms are logically ORed using OR gate. In case of POS expression we use OR–AND implementation. Here, we implement all sum terms using OR gates in the first level and all sum terms are logically ANDed using AND gate, to get product of sum, in the second level. We know that, the logic gates are available in the integrated circuit (IC) packages. When we implement logic circuit using basic gates, we require ICs for AND, OR and NOT gates.

Many times it may happen that all gates from the IC packages are not required to build the circuit and thus remaining gates are unused. Consider a combinational circuit which requires two 2–input AND gates and one 2–input OR gate as shown in Fig. 2.7.3. To implement such a circuit  we require IC 7408 (Four 2–input AND gates) and IC 7432 (Four 2–input OR gate). When we use these two ICs we find that two 2–input AND gates are unused and three 2–input OR gates are unused. Thus, the utility factor is very poor. This utility factor can be increased by using universal gates to implement logic functions.

Examples with Solutions

Example: 1

Implement the expression.

1) AB + BCD + EFGH

2) (A + B) (C + D + E) (F + G + H + I) with logic gates.

Solution :

Example: 2

Write the Boolean expression for the output of the system shown in Fig. 2.7.6 below :

Solution :

Example: 3

Draw the logic diagram for X = AB + B'C.

Solution :

Example: 4

Implement the Boolean expression using gates.

X = (AB + C') D + E.

Solution :

Example: 5

Implement the following expressions using basic gates.

Solution :