# Full Subtractor Circuit Construction using Logic Gates

Generally, the full subtractor is one of the most used and essential combinational logic circuits. It is a basic electronic device, used to perform subtraction of two binary numbers. In the earlier article, already we have given the basic theory of half adder & a full adder which uses the binary digits for the computation. Likewise, the full-subtractor uses binary digits like 0,1 for the subtraction. The circuit of full subtractor can be built with logic gates such as OR, Ex-OR, NAND gate. The inputs of this subtractor are A, B, Bin and outputs are D, Bout.

This article gives full-subtractor theory idea which comprises the premises like what is a subtractor, full subtractor design with logic gates, truth table, etc. This article is useful for engineering students who can go through these topics in HDL Practical lab.

## What is Full Subtractor?

Full subtractor is an electronic device or logic circuit which performs subtraction of two binary digits. It is a combinational logic circuit used in digital electronics. Many combinational circuits are available in integrated circuit technology namely adders, encoders, decoders and multiplexers. In this article, we are going to discuss full subtractor construction using half subtractor and also the terms like truth table. A full subtractor is formed by two half subtractors, which involves three inputs such as minuend, subtrahend and borrow, borrow bit among the inputs is obtained from subtraction of two binary digits and is subtracted from next higher order pair of bits, outputs as difference and borrow.

## Full Subtractor Block Diagram

The foremost disadvantage of the half subtractor is, we cannot make a Borrow bit in this subtractor. Whereas in full subtractor design, actually we can make a Borrow bit in the circuit & can subtract with remaining two i/ps. Here A is minuend, B is subtrahend & Bin is borrow in. The outputs are Difference (Diff) & Bout (Borrow out). The complete subtractor circuit can obtain by using two half subtractors with an extra OR gate. ### Full Subtractor Circuit Diagram with Logic Gates

The circuit diagram of full subtractor using basic gates is shown in the following block diagram. This circuit can be done with two half-Subtractor circuits.

In the initial half-Subtractor circuit, the binary inputs are A and B. As we have discussed in the previous half-Subtractor article, it will generate two outputs namely difference (Diff) & Borrow.

The difference o/p of the left subtractor is given to the Left half-Subtractor circuit’s. Diff output is further provided to the input of the right half Subtractor circuit. We offered the Borrow in bit across the other i/p of next half subtractor circuit. Once more it will give Diff out as well as Borrow out the bit. The final output of this subtractor is Diff output.

On the other hand, the Borrow out of both the half Subtractor circuits is connected to OR logic gate. Later than giving out OR logic for two output bits of the subtractor, we acquire the final Borrow out of the subtractor. The last Borrow out to signify the MSB (a most significant bit).

If we observe the internal circuit of the full Subtractor, we can see two Half Subtractors with NAND gate and XOR gate with an extra OR gate.

### Full Subtractor Truth Table

This subtractor circuit executes a subtraction between two bits, which has 3- inputs (A, B and Bin) and two outputs (D and Bout). Here the inputs indicate minuend, subtrahend, & previous borrow, whereas the two outputs are denoted as borrow o/p and difference. The following image shows the truth table of full-subtractor.

 Inputs Outputs Minuend (A) Subtrahend (B) Borrow (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

### Full Subtractor K-Map

The simplification of the K-map for the above difference and borrow is shown below. The full subtractor equations for the difference as well as Bin are mentioned below. The full subtractor expression for Difference is,

D = A’B’Bin + AB’Bin’+ A’BBin’ + ABBin

The full-subtractor expression for Borrow is,

Bout = A’Bin + A’B + BBin

### Applications of Full Subtractor

Some of the applications of full-subtractor include the following

• These are generally employed for ALU (Arithmetic logic unit) in computers to subtract as CPU & GPU for the applications of graphics to decrease the circuit difficulty.
• Subtractors are mostly used for performing arithmetical functions like subtraction, in electronic calculators as well as digital devices.
• These are also applicable for different microcontrollers for arithmetic subtraction, timers, and program counter (PC)
• Subtractors are used in processors to compute tables, address, etc.
• It is also useful for DSP and networking based systems.

From the above information, by evaluating the adder, full subtractor using two half subtractor circuits, and its tabular forms, one can notice that Dout in the full-subtractor is accurately similar to the Sout of the full-adder. The only variation is that A (input variable) is complemented in the full-subtractor. Thus, it is achievable to change the full-adder circuit into full-subtractor by just complementing the i/p A before it is given to the logic gates to generate the last borrow-bit output (Bout).

By using any full subtractor logic circuit, full subtractor using NAND gates and full subtractor using nor gates can be implemented, since both the NAND and NOR gates are treated as universal gates. Here is a question for you, what is the difference between half subtractor and full subtractor?