Adders

Binary Adders

Arithmetic circuit Addition Subtraction Division Multiplication

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

One bit in sum

Two bit in sum

Half Adder

A combinational circuit that performs the addition of two bits.

Two inputs and two outputs.

Augend and Addend

Sum and Carry

Truth table of Half Adder

X Y C S0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 0

Circuit Diagram

Full Adder

A combinational circuit that performs the addition of three input bits.

Three inputs and two outputs.

Sum and Carry

Truth table of Full Adder

X Y Z C S0 0 0 0 00 0 1 0 10 1 0 0 10 1 1 1 01 0 0 0 11 0 1 1 01 1 0 1 01 1 1 1 1

Circuit Diagram

Binary Ripple Carry Adder

Adders connected in cascade. Carry output from one full adder

connected to carry input of next full adder.

Binary Ripple Carry Adder

Input carry 0110 A 1011 B 0011 Sum 1110 Output carry 0011

Input carry in the least significant position is 0.

Simple in concept. Long circuit delay.

Many gates in the carry path.

Why a Carry Lookahead Adder?

Practical design with reduced delay.

For a n- bit ripple carry adder The longest delay path is 2n + 2. 16 – bit ripple carry adder - delay is 34

gate delays

Carry Lookahead Adder

Designed by a transformation of the ripple carry adder design in which the carry logic over fixed groups of bits of the adder is reduced to two-level logic.

Design

OR gate and one of the AND gates are removed to form each of the full adders to form the ripple carry adder.

Separate the parts of full adders not involving the carry propagation path from those containing the path.

First part of each full adder partial full adder - PFA

Two outputs Pi and Gi

From each PFA to ripple carry path

One input Ci

From the carry path to each PFA

Pi = Ai XOR Bi - Propagate function Gi = Ai . Bi - Generate function

Whenever Pi = 1 Incoming carry is propagated through bit

position from Ci+1.

Whenever Pi = 0 carry propagation through bit position is

blocked.

Whenever Gi = 1 Carry output from the position is 1. Regardless of value of Pi.

A Carry has been generated. Whenever Gi = 0

carry is not generated. Ci+1 is 0.

Ci is also 0.

Generate and propagate functions correspond exactly to the half adder.

Essential in controling the values in ripple carry path.

PFA generates sum function by XOR of incoming carry, Ci and propagate function, Pi.