Combinational Logic Circuits

Adders

Objectives

• Discuss and design a half adder and create its truth table
• Implement a full adder using half adder
• Design a truth table for a full adder

Gates and Integrated Circuits

• Gates (AND gate, OR gate, Not gate) are not sold individually
• They are sold in units called integrated circuits (ICs).
• A chip is a small electronic device consisting of the necessary electronic components (transistors, resistors, and capacitors) to implement various gates.

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�Logic circuits�

• Digital logic chips are combined to produce circuits.
• Logic circuits can be categorized as either
• combinational logic
• or sequential logic

This lesson covers combinational logic.

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�Combinational Logic Circuit�

• Combining a number of basic logic gates in a larger circuit to produce more complex logical operations
• A combinational logic circuit consists of logic gates whose outputs at any time are determined directly from the present combination of inputs without regard to previous inputs
• Combinational circuit is a circuit in which we combine the different gates in the circuit, for example encoder, decoder, multiplexer and demultiplexer. 
• Combinational logic is used to build circuits that contain basic Boolean operators, inputs, and outputs. In a combinational circuit, an output is always based entirely on the given inputs.

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Combinational Logic Circuit

• Recall that Boolean algebra allows us to analyze and design digital circuits.
• The output of a combinational circuit is a function of its inputs, and the output is uniquely determined by the values of the inputs at any given time
• The following are some of the combinational circuits that we will discuss:

Classification of Combinational Logic

• There are three main types of combinational logic circuits:
1. Arithmetic and logical combinational circuits: These include Adders, Subtractors, Multipliers, Comparators.
2. Data handling combinational circuits – These include Multiplexers, Demultiplexers, priority encoders, decoders.
3. Code converting combinational circuits – These include Binary to Gray, Gray to Binary, Binary to Excess 3, seven-segment, etc.

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Adders

• An adder is a digital circuit that performs addition of numbers.
• In many computers and other processors, adders are used in the arithmetic logic units or ALU.
• Additionally, adders are used in different parts of the processor, such as in calculating addresses, table indices, incrementing and decrementing, and similar operations.

Types of Adders

• There are two main types of adders:
1. half adders
2. full adders.

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• Half adders can add two binary digits or bits.
• Full adders can add three binary digits or bits.

Review on Binary Addition

• Binary addition is much like our normal everyday addition (decimal addition), except that it carries on a value of 2 instead of a value of 10.
• There are four rules of binary addition:�0 + 0 = 0�0 + 1 = 1�1 + 0 = 1�1 + 1 = 10 (which is 0 carry 1)

Example 1

• 1 + 11 = ?
• Solution

1

+ 11

If we take the first column from the right, we get the binary addition of 1 and 1, which is: 1 + 1 = 10 = sum is 0 and carry 1

The rightmost digit of our answer is therefore 0.

• The second column from the right becomes: 0 + 1 + 1 (from the carry). In binary addition: 0 + 1 + 1 = 10 = 0 carry 1
• The second rightmost digit is a 0 and a 1 is carried to the next column. The next column doesn't exist (there are no numbers), therefore the 1 drops into the next slot of the answer. So our answer is: 1 0 0

Class Activity

• 1010 + 11 = ?
• 100101 + 10101 = ?

�Half-adder�

• A half-adder is a very simple combinational logic circuit with two inputs and two outputs
• The half-adder only adds two single bits together
• The half adder adds two single binary digits A and B. It has two outputs, sum (S) and carry (C)
• Consider adding two binary digits together: The three things to remember when adding binary digits are:
• 0 + 0 = 0
• 0 + 1 = 1 + 0 = 1
• 1 + 1 = 10.
• In the above there is a sum and carry at the outputs. Sum is an XOR. The Carry output is equivalent to that of an AND gate

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Half-adder

• The simplest half-adder design incorporates an XOR gate for Sum and an AND gate for Carry.

�Truth Table for a Half-Adder�

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�Full-adder�

• Full-adder is composed of two half-adders and an OR gate.
• The full-adder is a three input and two output combinational circuit.
• The first two inputs are A and B and the third input is an input carry as C-IN.
• The output carry is designated as C-OUT and the normal output is designated as S which is SUM.
• A full adder adds binary numbers and accounts for values carried in as well as out.

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�A Truth Table for a Full-Adder�

�A Logic Diagram for a Full-Adder�

Adders Summary

• Adders are very important circuits—a computer would not be very useful if it could not add numbers

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Reading

• Hennessy and Patterson Chapter 8.3 (Appendix B)

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