Basic Digital Techniques & Applications - PART 5

07:11:00 Nandan Bhalwankar 0 Comments

Combinational Logic Circuits

Combinational Logic is a type of digital logic which is implemented by Boolean circuits, where the output is a pure function of the present input only.Hence it is also known as time-independent / history-independent logic.  We all know personal computer which we use daily has Arithmetic and Logic Unit (ALU) for performing mathematical calculations. This is also constructed using combinational logic circuits. Examples of these circuits are

a) Half Adder

b) Full Adder

c) Half Subtractors

d) Full Substractors

e) Multiplexers

f) De-multiplexers

g) Encoder

h) Decoder

Let’s have a look at these examples.

a) Half Adder:

As the name indicates, it is used for addition of numbers. The half adder adds two single binary digits A and B. It has two outputs, sum (S) and carry (C).Since the circuit adds the two digits, it is called as ‘Half Adder’ circuit.  

Truth table of Half Adder circuit:

If we look at the truth table, for the first three cases, the addition looks obvious. But for the last case, when A=1 and B=1 as well, the sum =0 and carry is 1.

b) Full Adder:

The full adder adds the three one bit binary numbers A, B and C and produces Sum and Carry as the output. The full adder is basically a combination of adders.

Truth table of Full Adder circuit:

If we see the first four records in this truth table, it looks very similar to Half Adder circuit. The sum of row numbers 5,6,7 is but obvious. For the last row,addition and carry is as shown below:

   A=1

+ B=1

__________________

Sum=0   (Carry=1)

+C=1

__________________

Final Sum=1  (Carry=1)  [See the final row]

K-map simplification of Full Adder circuit:

Let’s have a look at the K-map simplification of Full Adder circuit. Input variables are mentioned on the top left. Their possible combinations for input digits 0 and 1 form a K-map. Please note the position of input combination 10 and 11. This is all about positioning. The output ‘sum’ and ‘carry’ is mentioned inside the K-map.

Now, based on ‘true’ values of output, a quad (a group of 4 logical high digits), pair (a group of 2 logical high digits) or singles (a single digit) are formed.As shown in above diagram, for carry, 3 pairs are formed. While for Sum, four singles are formed. Priority is always given to quads and pairs as compared to singles.

The Logical expression is formed based on these quads, pairs and singles. The expression is in SOP form, for ‘Sum’ K-map, the single formed when A=1 and B=C=0 yields A. For digit 0, complement is written.

Logic Diagram of Full Adder circuit:

As shown in the diagram above, all the products are formed with logical AND gate. For sum, logical OR gate is used.