DKT 122/3
DIGITAL SYSTEM 1
WEEK #10
FUNCTIONS OF COMBINATIONAL LOGIC
(ADDERS)
Topic Outlines
ALL ABOUT ADDERS..
 Basic Adders
 Half-adder
 Full-adder
 Parallel Binary Adders
 Ripple Carry Adders
 Carry Look-Ahead Adders
Something To Share About Adders..
By Lewis Carroll
"Can you do addition?" the White Queen
asked.
"What's one and one and one and one and
one and one and one and one and one and
one?"
"I don't know," said Alice. "I lost count."
Through the Looking Glass.
Basic Adders
Half-adder
The half-adder (HA) accepts two binary digits on its
inputs, A & B and produces two binary digits on its
outputs, a sum bit,  and a carry bit, Cout
A
Half
B
Adder

Cout
Half-adder block diagram
Half-adder logic symbol
Basic Adders
Half Adder Truth Table
Half-adder
Basic Rule for Binary Addition
0+0=0
0+1=1
1+0=1
1 + 1 = 10
A
B
Outputs
Inputs
Sum,
Cout
A
0
0
1
1
B
0
1
0
1

0
1
1
0
 ( A, B)   m(1,2)
  AB  AB
 A  B
C ( A, B)   m(3)
out
Logic circuit for half-adder
Cout
0
0
0
1
Cout  AB
Basic Adders
Full-adder
Full adder (FA) accepts two input bits, A & B and
an input carry, Cin and generates a sum output, 
and an output carry, Cout
A
Full
B
Adder
Cin

Cout
Full-adder block diagram
Full-adder logic symbol
Basic Adders
Full Adder Truth Table
Full-adder
Outputs
Inputs
Basic Rule for Binary Addition
0+0=0
0+1=1
1+0=1
1 + 1 = 10
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
Cin
0
1
0
1
0
1
0
1
Cout ( A, B, Ci n )   m(3,5,6,7)
Cout  ABCin  ABCin  ABCin  ABCin
 ( A, B,C
Sum,   ABC
)   m(1,2,4,7)
 ABCin  ABCin ABC in
in
in
Cout
0
0
0
1
0
1
1
1

0
1
1
0
1
0
0
1
Basic Adders
Full-adder
Based on the previous truthtable, simplify the sum & carry
equation using K-map method:
Sum 
AB
00
Cin
01
11
10
Sum, Σ= ?
0
1
Carry Cout
AB
00
Cin
0
1
01
11
10
Carry, Cout = ?
Basic Adders
Remember that Boolean operation for half-adder:
 A  B
Cout  AB
For full-adder:
Initially, it is known from the Truth-Table of Full-adder that
Sum,   ABCin  ABCin  ABCin ABC in
 Cin ( AB  AB)  Cin ( AB  AB)
 Cin ( A  B)  Cin ( A  B)
 A  B  Cin
For Cout,
Cout  ABCin  ABCin  ABCin  ABCin
Cout
 AB(Cin  Cin )  Cin ( AB  AB)
 AB  Cin ( A  B)
Basic Adders
Full-adder
Logic circuit for full-adder
Arrangement of two half-adders to form a full-adder
Parallel Binary Adders
 To add 2 binary numbers, a full adder is required for each bit
in the numbers. So, for:
 2-bit numbers -> 2 adders are needed;
 4-bit numbers -> 4 adders are needed; & so on..
 For the LSB position, can use either a half-adder;
or full-adder (with carry input being made 0 (grounded))
Basic 2-bit parallel
adder using 2 full adder
Parallel Binary Adders
C4 is the output carry of the MSB adder
C0 is the input carry to the LSB adder
C0
C4
Σ3
Σ2
Σ1
Σ0
1 (LSB) … 4 (MSB) are the sum outputs
Basic 4-bit parallel adder using 4 full-adder
Types of Parallel Adders
 2 types of parallel adders
 ripple carry (RC) adder
 carry look-ahead (CLA) adder
 Differs in terms of how the internal carries from
stage to stage are handled
 Externally, both types of adders are the same in
term of inputs and outputs
 The difference is the speed at which they can add
numbers  CLA is faster than RC
Ripple Carry Adders
4-bit parallel ripple carry adder (showing “worst-case” carry
propagation delays)
Carry Look-Ahead Adders
Conditions for carry generation & carry propagation
Carry Look-Ahead Adders
Carry generation & carry propagation in terms of the input bits
to a 4-bit adder.
Carry Look-Ahead Adders
Logic diagram for a 4-stage look-ahead carry adder