ECE 301 – Digital Electronics

Single-bit Adder Circuitsand

Multi-bit Adder Circuits

(Lecture #10)

ECE 301 - Digital Electronics 2

The Half Adder (HA)

Single-bit Adder Circuits

ECE 301 - Digital Electronics 3

Binary Addition

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

Sum Carry Sum

ECE 301 - Digital Electronics 4

The Half Adder

ECE 301 - Digital Electronics 5

The Half Adder

ECE 301 - Digital Electronics 6

The Full Adder (FA)

Single-bit Adder Circuits

ECE 301 - Digital Electronics 7

Binary Addition

0 0 0 00 0 1 1

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

Carry-out Sum

1 1 1 10 0 1 1

+ 0 + 1 + 0 + 1 1 10 10 11

Carry-in

ECE 301 - Digital Electronics 8

The Full Adder

Cin Cout

ECE 301 - Digital Electronics 9

The Full Adder

Cin

Cin

Cin

Cin

S Cout

S = X xor Y xor Cin

Cout = X.Y + X.Cin + Y.Cin

ECE 301 - Digital Electronics 10

The Full Adder

X

Y

Cin

S

Cout

ECE 301 - Digital Electronics 11

The Full Adder

Half Adder Half Adder

Cin

Cin

Cin

Cin + xy

ECE 301 - Digital Electronics 12

Multi-bit Adder Circuits

ECE 301 - Digital Electronics 13

Implementations of Multi-bit Adders:

1. Ripple Carry Adder2. Carry Lookahead Adder

ECE 301 - Digital Electronics 14

Ripple Carry Adder

Multi-bit Adder Circuits

ECE 301 - Digital Electronics 15

Ripple Carry Adder

1 0 1 0

1 0 0 1+

1 Carry-in

0 1 0 01Carry-out

11

Carry ripples from one column to the next

ECE 301 - Digital Electronics 16

Ripple Carry Adder

Carry ripples from one stage to the next

Carry-inCarry-out

ECE 301 - Digital Electronics 17

Ripple Carry Adder

n-bit Ripple Carry Adder Composed of n 1-bit Full Adders Carries ripple from LSB stage to MSB stage

Delay ~ (n)*(delay of single FA stage) Area required is linear in n

4-bit Ripple Carry Adder Composed of 4 1-bit Full Adders

ECE 301 - Digital Electronics 18

The Ripple Carry Adder is slow!

Why?

How can the speed of the adder be increased?

ECE 301 - Digital Electronics 19

Increasing the speed of the Adder

Method A: Include all inputs and outputs in the design

Inputs = Xi, Yi, Cin,i; Outputs = Si, Cout,i

1-bit 3 inputs 2 outputs 2-bit 5 inputs 3 outputs 4-bit 9 inputs 5 outputs

n-bit 2n+1 inputs n+1 outputs

Large number of operands, but only 2 logic levels Increase in speed Increase in area required

decrease propagation delay

increase # of logic gates

Use Truth Table and K-Map to derive logic functions

ECE 301 - Digital Electronics 20

Increasing the speed of the Adder

Method B: Manipulate the Boolean Algebra

(results in the design of the Carry Lookahead Adder)

ECE 301 - Digital Electronics 21

Carry Lookahead Adder

Multi-bit Adder Circuits