Binary Subtractor

Binary Subtractor

• The subtraction consists of four possible elementary operations, namely,

0 – 0 = 0

0 – 1 = 1 with 1 borrow

1 – 0 = 1

1 – 1 = 0

• In all operations, each subtrahend bit is subtracted from the minuend bit. In case of second operation the minuend bit is smaller than the subtrahend bit, hence 1 is borrowed. Just as there are half and full–adders, there are half and full–subtractors.

1. Half Subtractor

• A half–subtractor is a combinational circuit that subtracts two–bits and produces their difference. It also has an output to specify if a 1 has been borrowed. Let us designate minuend bit as A and the subtrahend bit as B. The result of operation A – B for all possible values of A and B is tabulated in Table 4.4.1.

• As shown in Table 4.4.1, half–subtractor has two input variables and two output variables. The Boolean expression for the outputs of half–subtractor can be determined as follows.

K–map simplification for half–subtractor

Logic diagram

Limitations of half–subtractor :

In multidigit subtraction, we have to subtract two bits along with the borrow of the previous digit subtraction. Effectively such subtraction requires subtraction of three bits. This is not possible with half–subtractor.

Example: 1

Draw half subtractor using NAND gates.

Solution :

For half subtractor :

Implementation :

2. Full–Subtractor

• A full–subtractor is a combinational circuit that performs a subtraction between two bits, taking into account borrow of the lower significant stage. This circuit has three inputs and two outputs. The three inputs are A, B and B_in denote the minuend, subtrahend and previous borrow, respectively. The two outputs, D and B_out, represent the difference and output borrow, respectively. Table 4.4.2 shows the truth table for full–subtractor.

K–map simplification of D and B_out

B_out = ĀB_in+ĀB+BB_in

Logic diagram

• The Boolean function for D (difference) can be further simplified as follows :

With this simplified Boolean function circuit for full–subtractor can be implemented as shown in Fig. 4.4.6.

A full subtractor can also be implemented with two half–subtractors and one OR gate, as shown in Fig. 4.4.7. The difference output from the second half–subtractor is the exclusive–OR of B_in and the output of the first half–subtractor, which is same as difference output of full–subtractor.

The borrow output for circuit shown in Fig. 4.4.6 can be given as,

B_out = ĀBin + ĀB + BB_in

This Boolean function is same as borrow out of the full–subtractor. Therefore, we can implement full–subtractor using two half–subtractors and OR gate.

Review Questions

1. Define half subtractor and full subtractor.

2. Design a half–subtractor combinational circuit to produce the outputs. Difference and borrow.

3. Explain the operation of a half subtractor with the help of logic diagram and truth table.

4. Write the logic expressions for the difference and borrow of a half subtractor.

5. Realize full–subtractor using K–map.

6. Write an expression for borrow and difference in a full subtractor circuit.

7. Design full–subtractor circuit and draw necessary truth tables.

8. Explain how full subtractor can be designed by using two half subtractor circuits. Draw the circuit diagram.