Combinational Logic

By : Ali Mustafa

Contents

• Adder

• Subtractor

• Multiplier

• Comparator

• Decoder

• Encoder

• Multiplexer

How to Analyze any combinational circuit like this?

Analysis Procedure

3.Repeat step 2 until the outputs of the circuit are obtained in terms of the input variables.

1.Label all gate outputs that are a function of input variables.

2.Label the gates that are a function of input variables and previous labeled gates with different arbitrary symbols.

• To obtain the output Boolean functions from a logic diagram:

Analysis Procedure

Analysis Procedure

Analysis Procedure

Analysis Procedure• Derivation of Truth Table

Analysis Procedure

Analysis Procedure

Summary :How to design a combinational circuit?

1. Determine the required number of inputs and outputs.

2.Derive the Truth Table.

3.Obtain the simplified Boolean functions.

4.Draw the logic diagram.

Example

• Design a combinational circuit with three inputs and one output.

• The output must equal “1” when the inputs are less than three and “0” otherwise. [use only NAND gates]

Example (cont.)

1-The system have three inputs and one output

Example (cont.)

2- Derive the truth table

Example (cont.)

Circuits to be Implemented

• Arithmetic circuits

–Half Adder

–Full Adder

–Binary Adder/Subtractor

–Binary Multiplier

• Magnitude Comparator

What's the example of Arithmetic Circuit?

One of the famous Digital Logic Circuits is the calculator.

Arithmetic circuits

• An arithmetic circuit is a combinational circuit that performs arithmetic operations such as addition, subtraction, multiplication and division with binary numbers or with decimal numbers in a binary code.

• A combinational circuit that performs the addition of two bits is called a “Half Adder”.

Half adder• It is required to design a combinational circuit

that adds two binary numbers and produce the output in the form of two bits sum and carry

Solution1- We need to determine the inputs and output of the system and give letters for all of them our system has two inputs (X, Y) and two outputs (S, C)

Half adder

2-Derive the truth table according to the given relation between outputs and inputs

– In the half adder block the output equals the sum of two binary inputs

Half adder (cont.)3- Obtain the simplified Boolean functions for

each output as a function of the input variables using K-map

4- Draw the logic diagram

Full Adder• It is required to add three binary numbers

Solution

1. From the specifications of the circuit, determine the required number of inputs and outputs and assign a letter (symbol) to each.

Full Adder (cont.)2. Derive the truth table according to the given

relation between outputs and inputs

Full Adder (cont.) 3- Obtain the simplified Boolean functions for

each output as a function of the input variables using K-map

Full Adder (cont.)

4- Draw the logic diagram

4-Bit Binary Adder (Ripple Carry Adder)

Ripple Carry Adder

Binary Subtractor

• The subtraction of binary number can be done most conveniently by means of complements

• The subtraction “ A-B “ is done by taking the 2’s complement of “ B “ and adding it to “ A “.– The 2’s complement can be obtained by taking the

1’s complement and adding “1” to the least significant bit.

– The 1’s complement can be implemented easily with inverter circuit and we can add “1” to the sum by making the initial input carry of the parallel adder equal to “1”.

Adder/Subtractor

SubtractorA B D B

0 0 0 0

0 1 1 1

1 0 1 0

1 1 0 0

A B C D B

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

HALF SUBTRACTOR FULL SUBTRACTOR

Subtractor

• HALF SUBTRACTOR

• D = A’B + AB’

• B = A’B

• FULL SUBTRACTOR

• D = A’B’C + A’BC’ + AB’C’ + ABC

• B = A’C + A’B + BC

4- Bit Subtractor

Assignment # 3

• Design 4-Bit Adder - Subtractor

4-Bit Adder - Subtractor

Binary multiplier

Self Study

• 3 x 4 Bit Multiplier

Magnitude Comparator

• It is required to design a circuit to compare between two inputs A={A1,A0} and B={B1,B0} both inputs consists of two binary bits the circuit has three outputs Greater, Less, Equal

Magnitude Comparator (cont.)

1. Determine the required number of inputs and output.

Magnitude Comparator (cont.)

2. Derive the Truth Table that defines the required relationship between inputs and outputs.

Magnitude Comparator (cont.)3- Get the simplified logic function of the

outputs using k-map

Magnitude Comparator (cont.)4- Draw the circuit

4 – Bit Comparator

4 – Bit Comparator