Digital Electronics

Adders

HALF ADDER

Half Adder - Definition:

Half adder,

  • is an one-bit binary adder.
  • is a digital circuit that adds two $1$-bit binary numbers (A and B) and gives sum and carry as output.

$$\{ Carry, Sum\} =(A+B)$$

Adding two $1$ bit numbers:

$$\{ Carry, Sum\} =(A+B)$$$$\{ 0, 0\} =(0+0)$$$$\{ 0, 1\} =(0+1)$$$$\{ 0, 1\} =(1+0)$$$$\{ 1, 0\} =(1+1)$$

Truth table:

A
A
B
B
Carry
Carry
Sum
Sum
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
1
0
0
Input
Input
Output
OutputText is not SVG - cannot display

Logical Expression:

Logical expression for the outputs sum and carry can be found using K-map,

Sum:

$$Sum=A\overline{B}+\overline{A}B$$$$Sum=A\oplus B$$

Carry:

$$Carry=AB$$

Circuit Diagram:

Using the logical expression for sum and carry, we can implement half adder using logic gates,$$Sum=A\oplus B$$$$Carry=AB$$ Logical expression for Sum can be implemented using an EXOR gate and Carry can be implemented using an AND gate.

A
A
B
B
Sum
Sum
Carry
CarryText is not SVG - cannot display

Doubt!!!

Can we connect two half adders serially, in order to make a $2$-bit adder??

NO

Second stage half adder can add input bits A and B, but not the incoming carry from first stage.

That’s why this adder is called as half-adder and the adder that can add incoming carry is termed as full-adder.