Adder Circuits (Digital Electronics) – Half and Full adder logic diagrams

#Technical studies – Study 20

Introduction

Adders in digital system are the combinational logic circuits that perform an arithmetic  operation called addition. The binary addition rules are –

0+0=0, 0+1=1, 1+0=1, 1+1=10

In the four rules stated above, the first three rules add two bits and the result is a single bit. But, in last rule two bits are added and the result is also two bits. The single digit result is called as sum. When the result is more than one digit the LSB is called sum and the MSB is called carry.

A combinational logic circuit that performs the addition of two numbers is called a half adder.

 Similarly, a combinational logic circuit that performs the addition of three numbers is called full adder.

Half adder:

Half adder is a digital logic circuit that takes two bits in adds them and provides two outputs one sum and one carry.

According to the truth table the sum output is similar to the output of X-OR operation. Therefore

S= =A⊕B

The carry output is similar to the output of AND operation. Therefore

C = AB

So the logic circuit for the half adder can be designed by using a X-OR gate and an AND gate.

Half adder using AOI (AND OR INVERTER) logic:

 Using AOI logic we can design the logic circuit for the half adder.

Half adder using NAND logic:

Half adder logic circuit can be designed using NAND logic.

     

     

     

     

And      

Half adder using NOR logic:

Half adder logic circuit can be designed using NOR logic.

   

   

   

   

 

And  

 Full adder:

Full adder is a digital logic circuit that takes three bits in adds them and provides two outputs one sum and one carry. Out of the three input bits two are the variables and the third one is the previous carry.

 According to the truth table the expression for sum and carry can be: –

$S = \overline{A} \overline{B} C_{in} + \overline{A} B {\overline{C}}_{in} + A \overline{B} {\overline{C}}_{in} + AB{C}_{in}$

    $= {\overline{C}}_{in} (\overline{A}B + A\overline{B}) + C_{in} (AB + \overline{A} \overline{B})$

     $= (A\oplus B) {\overline{C}}_{in} + \overline{(A\oplus B)} C_{in}$

     $= A\oplus B \oplus C_{in}$

 

And

$C_{out} = \overline{A}B{C}_{in} + A\overline{B}{C}_{in} + AB{\overline{C}}_{in} + AB{C}_{in}$

         $=C_{in} (\overline{A}B + A\overline{B}) + AB ({\overline{C}}_{in} +C_{in})$

         $= C_{in} (A\oplus B) + AB$

         

So we can design the logic circuit of full adder as below:

Full adder using AOI (AND OR INVERTER) logic:

 Design of full adder logic circuit Using AOI logic.

Full adder using NAND logic:

To design Full adder logic circuit using NAND logic we have to make the expression of full adder in NAND form.

The expression for sum and carry for the full adder logic circuit using NAND gates can be:

 

$S = A\oplus B\oplus C_{in}$

    $= (A\oplus B)\oplus C_{in}$ 

    $= \overline{(A\oplus B)} C_{in}+(A\oplus B) {\overline{C}}_{in}$

    $= \overline{(A\oplus B)} C_{in}+(A\oplus B)\left(\overline{A\oplus B}\right)+(A\oplus B){\overline{C}}_{in}+C_{in} {\overline{C}}_{in}$

    $= (A\oplus B)\left(\right(\overline{A\oplus B})+\overline{C_{in}})+C_{in}\left(\right(\overline{A\oplus B})+\overline{C_{in}})$

    $= (A\oplus B).\overline{(A\oplus B) C_{in}} + C_{in} . \overline{(A\oplus B) C_{in}}$

    $= \overline{\overline{(A\oplus B).\overline{(A\oplus B) C_{in}}}} + \overline{\overline{C_{in}.\overline{(A\mathit{\oplus }B) C_{in}}}}$  

    $=\overline{\overline{(A\oplus B).\overline{(A\oplus B) C_{in}}} . \overline{C_{in}.\overline{(A\oplus B) C_{in}}}}$

 

And 

$C_{out} = AB+(A\oplus B) C_{in}$

         $= \overline{\overline{AB}} + \overline{\overline{(A\oplus B) C_{in}}}$ 

         $= \overline{\overline{AB} . \overline{(A\oplus B) C_{in}}}$

Full adder using NOR logic:

We can design Full adder logic circuit using NOR logic.

To design full adder logic circuit using NOR gates the expressions are : 

 

$S= A\oplus B\oplus C_{in}$

   $= (A\oplus B) \oplus C_{in}$

   $= \overline{(A\oplus B)} C_{in}+(A\oplus B) {\overline{C}}_{in}$  

   $= \overline{(A\oplus B)} C_{in}+(A\oplus B)\left(\overline{A\oplus B}\right)+(A\oplus B){\overline{C}}_{in}+C_{in} {\overline{C}}_{in}$

   $= (A\oplus B)\left(\right(\overline{A\oplus B})+\overline{C_{in}})+C_{in}\left(\right(\overline{A\oplus B})+\overline{C_{in}})$

   $=( (A\oplus B) + C_{in}) .( \overline{(A\oplus B)} + \overline{C_{in}})$

    $= \overline{\overline{(A\oplus B) + C_{in}}} . \overline{\overline{\overline{(A\oplus B)} + \overline{C_{in}}}}$

    $= \overline{\overline{(A\oplus B) + C_{in}} + \overline{\overline{(A\oplus B)} + \overline{C_{in}}}}$

And 

$C_{out} = AB + C_{in} (A\oplus B)$

         $= \overline{\overline{AB}} + \overline{\overline{(A\oplus B) C_{in}}}$

         $= \overline{\overline{A} +\overline{B}} + \overline{\overline{A\oplus B} + \overline{C_{in}}}$

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