Half subtractor, full subtractor

Half Subtractor and Full Subtractor

In digital logic circuits, subtraction of binary numbers is performed using subtractor circuits. A half subtractor and a full subtractor are combinational circuits used for binary subtraction. Let's discuss each in detail.

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1. Half Subtractor

A half subtractor is a basic circuit that subtracts two single-bit binary numbers. It has two inputs and two outputs:

• Difference (D)
• Borrow (B)

Truth Table for Half Subtractor:

Input A	Input B	Difference (D)	Borrow (B)
0	0	0	0
0	1	1	1
1	0	1	0
1	1	0	0

Explanation:

• Difference (D) is the result of subtracting the second bit (B) from the first bit (A).
• Borrow (B) is generated when the bit being subtracted is larger than the bit being subtracted from (i.e., A < B).

Logic Equations:

• Difference (D) = A ⊕ B (A XOR B)
• Borrow (B) = A' · B (NOT A AND B)

Where:

• A' is the NOT of A (inverted A).

Circuit Design for Half Subtractor:

• The Difference is calculated using an XOR gate between inputs A and B.
• The Borrow is calculated using an AND gate between the inverted A (A') and B.

Diagram of Half Subtractor:

     A ───┐
           │
           │   XOR   ─── Difference
     B ───┘──────┐
                  │
                  │   AND    ─── Borrow
                  └──────────┘

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2. Full Subtractor

A full subtractor is a more advanced circuit that subtracts three binary bits. It has three inputs:

• A (Minuend)
• B (Subtrahend)
• Bin (Borrow input from the previous stage)

It produces two outputs:

• Difference (D)
• Borrow (Bout) (Borrow output for the next stage)

Truth Table for Full Subtractor:

A	B	Bin	Difference (D)	Borrow (Bout)
0	0	0	0	0
0	0	1	1	1
0	1	0	1	1
0	1	1	0	1
1	0	0	1	0
1	0	1	0	0
1	1	0	0	0
1	1	1	1	1

Explanation:

• Difference (D) is the result of subtracting the two bits (A and B) and considering the borrow input (Bin).
• Borrow (Bout) is generated when the bit being subtracted is larger than the bit being subtracted from, or when there is a carry from the previous subtraction.

Logic Equations:

• Difference (D) = A ⊕ B ⊕ Bin (A XOR B XOR Bin)
• Borrow (Bout) = (A' · B) + (A' · Bin) + (B · Bin)

Where:

• A' is the NOT of A (inverted A).
• Bin is the borrow input.

Circuit Design for Full Subtractor:

The full subtractor can be constructed using:

1. XOR gates to calculate the difference.
2. AND gates to calculate the borrow.
3. OR gate to combine the borrow outputs.

Diagram of Full Subtractor:

     A ───┐
           │
           │   XOR   ─── Difference
     B ───┘──────┐
                  │
           ┌──────┴──────┐
           │             │
           │             │
     Bin ───┘         AND ─── Borrow Out
                           │
                           OR
                           │
                           └─ Borrow out

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3. Difference Between Half Subtractor and Full Subtractor

Feature	Half Subtractor	Full Subtractor
Inputs	2 inputs (A and B)	3 inputs (A, B, and Borrow in)
Outputs	2 outputs (Difference and Borrow)	2 outputs (Difference and Borrow out)
Borrow Input	No borrow input	Has borrow input from the previous stage
Used in	Simple subtraction of two bits	Multi-bit subtraction (used in chained subtractors)
Complexity	Simple design with basic gates	More complex, requires XOR, AND, OR gates

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4. Use of Full Subtractor in Multi-Bit Subtraction

Just as full adders are used in multi-bit addition, full subtractors can be used to subtract multi-bit binary numbers. A full subtractor can be cascaded to perform multi-bit subtraction, where the borrow output from one full subtractor becomes the borrow input to the next full subtractor.

For example, to subtract two 4-bit binary numbers:

  A3 A2 A1 A0
- B3 B2 B1 B0
---------------
  D3 D2 D1 D0  (Difference)
Borrow Out

• Full subtractor 1 will subtract A0 - B0 considering the borrow input (Bin) from the previous stage.
• Full subtractor 2 will subtract A1 - B1 and the borrow output from the previous full subtractor, and so on.

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Conclusion

• Half Subtractor is a simple circuit that performs binary subtraction of two bits (A and B) and generates the difference and borrow outputs.
• Full Subtractor extends the half subtractor by adding a third input, the borrow input (from a previous stage), and provides the difference and borrow output, making it suitable for multi-bit subtraction.
• Full subtractors are useful when subtracting larger binary numbers, as they can be cascaded to handle multi-bit binary subtraction.