1. WEEK #10 FUNCTIONS OF COMBINATIONAL LOGIC (ADDERS)
DKT 122/3 DIGITAL SYSTEM 1
WEEK #10
FUNCTIONS OF COMBINATIONAL LOGIC (ADDERS)

2. Topic Outlines ALL ABOUT ADDERS.. Basic Adders Half-adder Full-adder
Parallel Binary Adders
Ripple Carry Adders
Carry Look-Ahead Adders

3. 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.

4. 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
Half
Adder A B 
Cout
Half-adder block diagram
Half-adder logic symbol

5. Basic Rule for Binary Addition
Basic Adders
Half-adder
Half Adder Truth Table
Inputs
Outputs A 1 B 1
Cout 1  1
Basic Rule for Binary Addition
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 A B
Sum,
Cout
Logic circuit for half-adder

6. 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
Adder  B
Cout
Cin
Full-adder block diagram
Full-adder logic symbol

7. Basic Rule for Binary Addition
Basic Adders
Full Adder Truth Table
Full-adder
Inputs
Outputs A 1 B 1
Cin 1
Cout 1  1
Basic Rule for Binary Addition
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10

8. Basic Adders Full-adder
Based on the previous truth-table, simplify the sum & carry equation using K-map method:
Sum, Σ= ?
Carry, Cout = ?
Sum  AB
Cin 1
Carry Cout AB
Cin 1

9. Basic Adders Remember that Boolean operation for half-adder:
For full-adder:
Initially, it is known from the Truth-Table of Full-adder that
For Cout,

10. Basic Adders Full-adder Logic circuit for full-adder
Arrangement of two half-adders to form a full-adder

11. 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

12. 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

13. 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

14. Ripple Carry Adders
4-bit parallel ripple carry adder (showing “worst-case” carry propagation delays)

15. Carry Look-Ahead Adders
Conditions for carry generation & carry propagation

16. Carry Look-Ahead Adders
Carry generation & carry propagation in terms of the input bits to a 4-bit adder.

17. Carry Look-Ahead Adders
END
Logic diagram for a 4-stage look-ahead carry adder