Overview
This lecture introduces the concept of the binary full adder, its necessity in digital circuits, and describes in detail how it operates, how to build its circuit, and how it enables multi-bit binary addition.
Half Adder Recap
- A half adder adds two 1-bit binary numbers (A and B).
- Produces two outputs: sum (ones place) and carry (overflow bit).
- Sum output is 1 if only one input is 1; carry output is 1 if both inputs are 1.
- Half adder is built using an XOR gate (for sum) and an AND gate (for carry).
Limitations of the Half Adder
- Only works for single-bit binary addition.
- Cannot handle multi-bit additions where there is a carry input from the previous column.
Introduction to the Full Adder
- A full adder adds three binary inputs: A, B, and carry-in (Cin).
- Outputs are sum and carry-out (Cout).
- Enables chaining to add multi-bit binary numbers.
Full Adder Truth Table and Logic
- The full adder truth table covers all possible combinations of A, B, and Cin.
- Sum output is 1 when the total number of ones in inputs is odd.
- Carry-out output is 1 when at least two of the inputs are 1.
Building the Full Adder Circuit
- First, a half adder is used for A and B.
- The sum from the first half adder and Cin are then input to a second half adder.
- Final sum output is from the second half adder's sum.
- Both half adder carry outputs are combined using an OR gate for the final carry out.
Chaining Adders for Multi-Bit Addition
- First stage (least significant bit): use a half adder or a full adder with Cin set to 0.
- Subsequent stages: use full adders, passing each carry-out to the next carry-in.
- This structure allows addition of binary numbers of any bit length.
Key Terms & Definitions
- Half Adder — Circuit that adds two single binary digits and outputs sum and carry.
- Full Adder — Circuit that adds three binary digits (A, B, Cin) and outputs sum and carry.
- Carry-in (Cin) — Input carry from a less significant bit's addition.
- Carry-out (Cout) — Output carry to the next more significant bit's addition.
- XOR Gate — Logic gate that outputs 1 when inputs differ.
- AND Gate — Logic gate that outputs 1 only if all inputs are 1.
- OR Gate — Logic gate that outputs 1 if at least one input is 1.
Action Items / Next Steps
- Review the full adder truth table and circuit diagram.
- Practice drawing and labeling a full adder using logic gates.
- Try adding multi-bit binary numbers using chained full adders.