2.2 Combinational Circuit

Combination Circuits

•      A combinational circuit or logic block is an interconnected set of gates and contains no memory whose computes the output at any time is a function only of the given current inputs.
•   Gates can be defined in three ways:

o   Truth Table 

For each of the 2n possible combinations of input signals, the binary value of each of the F output signals is listed.

o   Boolean equations

Each output signal is expressed as a Boolean function that consist possible combination of inputs.

                                o    Graphical symbols

 

 

The interconnected layout of gates is depicted.

Boolean Equation Forms

Any Boolean function can be implemented in electronic form as a network of gates.

For any given function, there are a number of alternative realizations.

All Boolean equation can be represented in two forms:

-          Sum of products (SOP)

o   Equivalent variables, ANDed together then ORed with other combination variables with the same output are converted from the combination of input values that produce 1s, 

o   Truth table is easier to be derived by SOP.

-          Product of sums (POS)

o   0s in sum terms (ORed variables) that produced by input combinations are ANDed together.

o   Convert input values that produce 0s into equivalent variables, ORed the variables, then ANDed with other ORed forms.

o   If more 1s produce in output function, POS usually used.

Example:

F = C’D’ + A’BCD

The truth table:

A	B	C	D	F
0	0	0	0	1
0	0	0	1	1
0	0	1	0	1
0	0	1	1	1
0	1	0	0	0
0	1	0	1	0
0	1	1	0	0
0	1	1	1	0
1	0	0	0	0
1	0	0	1	0
1	0	1	0	0
1	0	1	1	0
1	1	0	0	0
1	1	0	1	1
1	1	1	0	0
1	1	1	1	0

     Product term

     A’B’C’D’

     A’BC’D’

     A’BCD

     AB’C’D’

     ABC’D’

SOP expression:

F = (A’B’C’D’) + (A’B’C’D’)+(A’BCD)+(AB’C’D’)+(ABC’D’)

2．

F = (X+Y+Z)(X+Y+Z’)(Z+Y’+Z)(X’+Y+Z’)

The truth table:

 

X	Y	Z	F
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	1
1	0	0	1
1	0	1	0
1	1	0	1
1	1	1	1

Product term:

            X+Y+Z

            X+Y+Z’

            X+Y’+Z

            X’+Y+Z’

          

POS expression:

F = (X+Y+Z)(X+Y+Z’)(X+Y’+Z)(X’+Y+Z’)