What is Adder and how many types?

Table of Contents

Adder:

Adder is the main part of computer’s ALU (Arithmetic and Logic Unit). With the help of which we perform all types of Arithmetical and Logical operations. But ALU never works with decimal numbers, that’s why two or more binary numbers can be added with this type of logic circuit. Adder is binary summator. So you need to know the general rules of binary addition. For example 0+0=00, 0+1=01, 1+0=01, 1+1=11. In this case the sum is shown in two bits where the left bit is carry and the right bit is sum. Adder is of two types. Namely –

  1. Half Adder
  2. Full Adder

Half Adder:

Half Adder is the simplest combinational logic circuit, with the help of which we can add two binary bits (x and y). Sum and carry are calculated by adding two bits together with the help of half adder. Half Adder truth table is shown –

Half Adder Truth Table
Half Adder Truth Table

In this case the Boolean equations of sum and carry are –

Sum = x⊕y = xy + xy; Carry= x.y Therefore
Sum is x XOR y; Carry is x AND y
So the logic circuit for the logic gate of Half Adder is the combination of XOR gate for Sum (S) and AND gate for Carry (C).

From the truth table we get:

S (Sum) = 1 then,
When x = 0 and y = 0 from the truth table
∴ S=x.y [Remember: If x or y has value 0 (zero), bar ( ‾‾ ) sits on its head]
Again S (Sum) = 1,
When x = 1 and y = 0 from the truth table
∴ S = x.y
So, S = x.y + x.y [The two are OR (+).]
S(Sum) = x⊕y
Again, looking at carry or C, we find that C=1 if x=1 and y=1.
∴ C = x.y

HALF ADDER LOGIC CIRCUIT AND BLOCK DIAGRAM
HALF ADDER LOGIC CIRCUIT AND BLOCK DIAGRAM

Full Adder:

Full Adder is a combinational logic circuit that adds three or more bits. Like half adder, full adder and three bits are added together to calculate the sum and carry. Full Adder will have a carry bit (Ci called carry in) in addition to two bits x and y as input and S (Sum) and C0 carry out as output. Full Adder truth table is shown-

Full Adder Truth Table
Full Adder Truth Table

From the truth table we get:

S (Sum )= 1, when
x=0 y=0 Ci=1 [i.e xyCi]
x=0 y=1 Ci=0 [i.e xyCi]
x=1 y=0 Ci=0 [i.e xyCi]
x=1 y=1 Ci=1 [i.e xyCi]
i.e S= x yCi + xyCi + x yCi + xyCi
i.e C0 (Carry Out) = 1 when will,
x=0 y=1 Ci=1 [i.e xyCi]
x=1 y=0 Ci=1 [i.e xyCi]
x=1 y=1 Ci=0 [i.e xyCi]
x=1 y=1 Ci=1 [i.e xyCi]
∴ C0(Carry out) = xyCi + xyCi + xyCi + xyCi

From the truth table, S (Sum) and C0 (Carry out) are Boolean equations –
S= x yCi + xyCi + x yCi + xyCi
= x(yCi + yCi) + x(y Ci + yCi)
= x(y⊕Ci) + x(y⊕Ci)
=x⊕y⊕Ci [Sum = x XOR y XOR Ci ]
Again, C0= xyCi + xyCi +xyCi + xyCi
C0= x(yCi + yCi) + x(yCi + yCi)
= Ci(xy + xy) + xy (Ci + Ci)      [we know A + A = 1]
=Ci (x⊕y) + xy
Then we can make the logic circuit of Full Adder from Boolean equation is-

Logic Diagram of FULL Adder in Connection of Two HALF Adder
Logic Diagram of FULL Adder in Connection of Two HALF Adder

In fact, two half adders are connected consecutively and form a full adder with the help of an OR gate. The logic diagram of the adder is shown-

Full Adder Block Diagram
Full Adder Block Diagram

Frequently Asked Question (FAQs):

  • What is Adder?
  • Half Adder.
  • Full Adder.

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