Half Subtractor is a basic combinational circuit used in a modern digital computer for performing the subtraction of two binary bits. Half Subtractor has various applications in the domain of Digital Electronics. In this article, we will study Half Subtractor, its operations along with its circuit. So, let us start with the definition of Half Subtractor.
What is Half Subtractor?
A half subtractor is a digital logic circuit that performs the subtraction of two binary bits. It has two inputs, A and B, and two outputs, Difference (d) and Borrow (b). The output d is the difference between the two inputs, A and B, and output b represents a borrow, indicating that the subtraction of B from A requires borrowing from the next digit in a larger binary number.
A half subtractor can be used in combination with other half subtractors to build a full subtractor circuit, which can perform binary subtraction of larger binary numbers.
The block diagram for the half subtractor is shown below:
Let us make the truth table according to the behavior of a Half Subtractor on giving various different inputs.
Truth Table of Half Subtractor
The truth table for Half Subtractor is given as
From the truth table, we can see that the difference (d) output is equal to 1 if either A is 1 or B is 0. This means that the half subtractor will produce a 1 in the difference output if there is no borrow from the next lower order bit. If both A and B inputs are 1, then the half subtractor produces a 0 in the difference output and a 1 in the borrow out (b) output. This indicates that there is a borrow from the next lower order bit.
K-Map for Half Subtractor
After making the Truth Table for the Half Subtractor, let us now derive the Boolean Expression for both the outputs of Half Subtractor i.e., “d” and “b”.
K-Map is the official way for deriving the boolean expressions using the truth table for a particular digital circuit. Let us make the K-Map for the Half Subtractor.
For Difference (d):
For Borrow (b):
So by using the K-Map we get the Logical Boolean Expressions for the output as:
- d = A’B + AB”, which can be written as
d = A ⊕ B
- b = A’B
Construction of Half Subtractor
Since, we get the Boolean Expression, now we are constructing the Circuit Diagram for the Half Subtractor.
The Half Subtractor can be constructed using the following basic Logic Gates:
- 1 AND Gate
- 1 Ex-OR Gate
- 1 NOT Gate
Before making Circuit Diagram, it is good for us to learn the above-mentioned logic gates in brief:
- AND Gate:
An AND gate is a digital logic gate that outputs a 1 (high voltage) if and only if all its inputs are 1. Its Symbol is
The truth table for AND Gate is given below:
- Ex-OR Gate:
An XOR (Exclusive OR) gate is a digital logic gate that outputs a 1 if and only if exactly one of its inputs is 1. Its Symbol is
The truth table for Ex-OR Gate is given below:
- NOT Gate:
A NOT gate is a digital logic gate that outputs the opposite logic level of its single input, effectively inverting it. Its Symbol is
The truth table for NOT Gate is given below:
Circuit Diagram for Half Subtractor using Basic Gates:
The circuit diagram according to the above mentioned boolean expression is given below:
Half Subtractor using only NAND Gates:
We can also construct a Half Subtractor using only NAND Gates as shown in the circuit diagram given below:
Here, X and Y are inputs and D represents “difference” between bits and B denotes “borrow”.
Applications of Half Subtractor
A half subtractor is a combinational circuit that is used to perform the subtraction of two binary bits. It has several applications, including
- Digital subtraction – The half subtractor can be used to build larger digital circuits for binary subtraction.
- BCD subtraction – BCD (Binary-Coded Decimal) subtraction can be performed using half subtractors.
- Error detection – In digital communication systems, half subtractors can be used to detect errors in the transmission of binary data.
- Arithmetic logic unit (ALU) – The half subtractor can be used in the design of an ALU (Arithmetic Logic Unit) in a computer’s central processing unit (CPU).
- Other digital circuits – The half subtractor can be used in other digital circuits such as multiplexers, decoders, and comparators.
Limitations of Half Subtractor
A half subtractor has some limitations, which are listed below:
- Only handles subtraction of two single-bit binary numbers.
- Cannot perform addition or handle multi-bit numbers.
- Only gives a difference and a borrow output, not the sum and carry.
- Cannot handle negative numbers or perform signed subtraction.
In conclusion, a half subtractor is a digital circuit that performs the subtraction of two binary digits and provides two outputs, difference and borrow. It is a basic building block in digital circuits and is commonly used in applications such as arithmetic circuits, data processing, and computer memory. The half subtractor can be implemented using basic logic gates, such as XOR and AND gates, and its behavior can be described by a truth table. Understanding the working of a half subtractor is crucial for understanding more complex digital circuits and systems.
Frequently Asked Questions (FAQs)
Some Frequently Asked Questions on Half Subtractor are
Q1. How does a half subtractor work?
Ans. A half subtractor takes two binary digits, A and B, as inputs and produces two outputs, D and B. The difference (D) output is equal to A minus B, and the borrow (B) output indicates whether a borrow is required from the next significant digit during subtraction.
Q2. How is a half subtractor implemented using gates?
Ans. A half subtractor can be implemented using basic logic gates such as XOR and AND gates. The XOR gate provides the difference output, while the AND gate provides the borrow output.
Q3. What is the difference between a half subtractor and a full subtractor?
Ans. A half subtractor is a digital circuit that performs subtraction of two binary digits and provides two outputs, difference and borrow. A full subtractor, on the other hand, is a digital circuit that performs the subtraction of three binary digits and provides two outputs, difference and borrow. The full subtractor has an additional input, carry-in, to account for the carry from the previous subtraction.