Half Adders & Full Adders

What is a Half Adder Circuit?

• A basic digital circuit used in computation to perform the addition of two single-bit numbers
• It has two inputs, usually labelled as A and B
• It produces two outputs: Sum ($S$) and Carry out ($C_{out}$)

Half Adder Logic

The logic can be expressed using Boolean algebra:

• Sum ($S$): $A\underline{\vee }B$ (A XOR B)
• Carry ($C_{out}$): $A\wedge B$ (A AND B)

Truth Table

A	B	Carry ($C_{out}$)	Sum ($S$)
0	0	0	0
0	1	0	1
1	0	0	1
1	1	1	0
		$A\wedge B$	$A\underline{\vee }B$

Calculation Tip:

• Remember that you are simply adding the binary numbers represented by A and B
• $0+0=0$ (Sum 0, Carry 0)
• $0+1=1$ (Sum 1, Carry 0)
• $1+1=2$ (which is $10_2$, so Sum 0, Carry 1)

Circuit Diagram

A half adder consists of one XOR gate (to calculate the Sum) and one AND gate (to calculate the Carry).

*Half Adder Logic Gates*

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Full Adders

What is a Full Adder Circuit?

• Extends the capabilities of the half adder to handle the addition of three bits
• This allows for the inclusion of a carry bit from a previous addition operation
• It has three inputs: A, B, and an input carry ($C_{in}$)
• It produces two outputs: Sum ($S$) and Carry out ($C_{out}$)

Truth Table

A	B	$C_{in}$	Carry ($C_{out}$)	Sum ($S$)	Value ($A+B+C_{in}$)
0	0	0	0	0	0
0	0	1	0	1	1
0	1	0	0	1	1
0	1	1	1	0	2 ($10_2$)
1	0	0	0	1	1
1	0	1	1	0	2 ($10_2$)
1	1	0	1	0	2 ($10_2$)
1	1	1	1	1	3 ($11_2$)

Calculation Tip:

• The full adder adds three inputs: A, B and $C_{in}$
• The total sum can be 0, 1, 2 or 3
• Write the result as a 2-bit binary number in the output columns
• Example: If A=1, B=1, $C_{in}$=1, the total is 3. In binary, 3 is $11$, so $C_{out}$ is 1 and $S$ is 1

Circuit Diagram

A full adder is constructed by combining two half adders and an OR gate.

1. First Half Adder: Adds A and B to produce a partial sum and partial carry
2. Second Half Adder: Adds the partial sum to $C_{in}$
3. OR Gate: Checks if either half adder generated a carry bit to produce the final $C_{out}$

*Full Adder Logic Gates*

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Ripple Carry Adders

A single full adder can only add single binary digits. To add larger binary numbers (e.g. 4-bit or 8-bit numbers), we must connect multiple full adders together.

• This configuration is known as a Ripple Carry Adder
• To add two $n$-bit numbers, you need $n$ full adders
• The $C_{out}$ from one adder is connected directly to the $C_{in}$ of the subsequent adder
• This represents the fundamental arithmetic process inside a CPU’s Arithmetic Logic Unit (ALU)

<img src=“https://upload.wikimedia.org/wikipedia/commons/5/5d/4-bit_ripple_carry_adder.svg” alt=“Ripple Carry Adder” width:“500” /> Ripple Carry Logic

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Worked Example

Question: A half-adder circuit is shown in the diagram below.

Describe how this logic circuit can be adapted to add together two 4-bit binary numbers. [4 marks]

Answer:

• Step 1: The current circuit is a half adder, which only adds two bits. To add larger numbers, we first need to create a Full Adder
• Step 2: This is done by joining two half adders together with an OR gate
• Step 3: To add two 4-bit numbers, we need to connect 4 Full Adders together in a chain
• Step 4: This is achieved by taking the Carry Out ($C_{out}$) from the first adder and connecting it to the Carry In ($C_{in}$) of the next adder in the sequence (a Ripple Carry Adder)