## Combinational Circuits

**Combinational circuits:**

** **Digital logic circuits are basically categorized into two types

- Combinational circuits in which there are no feedback paths from outputs to inputs and there is no memory
- Sequential circuits in which feedback paths exist from outputs to inputs and they have memory

Block diagram: A combinational circuit is a connected arrangement of logic gates with a set of inputs and outputs. At any given time, the binary values of the outputs are a function of the binary combination of the inputs. The n binary input variables come from an external source, the m binary output variables go to an external destination and in between there is an interconnection logic gates. A combinational circuit transforms binary information from the given input data to the required output data.

Analysis: The analysis of a combinational circuit starts with a given logic circuit diagram and culminates with a set of Boolean functions or a truth table. If the digital circuit is accompanied by a verbal explanation of its function, the Boolean functions or the truth table is sufficient for verification. If the function of the circuit is under investigation, it is necessary to interpret the operation of the circuit from the derived Boolean functions or the truth table. The success of such investigation is enhanced if one has experience and familiarity with digital circuits

Design: The design of combinational circuits starts from the verbal outline of the problem and ends in a logic circuit diagram. The procedure involves the following steps

- The problem is stated
- The input and output variables are assigned letter symbols
- The truth tables that defines the relationship between inputs and outputs is derived
- The simplified Boolean functions for each output are obtained
- The logic diagram is drawn

**Half Adder:**

** ** The most basic digital arithmetic circuit is the addition of two binary digits. A combinational circuit that performs the arithmetic addition of two bits called a half adder. The input variables of a half adder called the augend and addend bits. The output variables the sum and carry. It is necessary to specify two output variables because the sum of 1+1 is binary 10, which has two digits. We assign symbols x and y to the two input variables and S for sum, and C for carry to the two output variables. The C output is 0 unless both inputs are 1. The S output represents the least significant bit of sum.

Logic diagram consists of an exclusive OR and an AND gate. A half adder logic module of an exclusive OR gate and an AND gate can be used to implement universal logic gates NAND and NOR

**Full Adder**:

A full adder is a combinational circuit that forms the arithmetic sum of three input bits. It consists of three inputs and two outputs. Two of the input variables denoted by x and y, represent the two significant bits to be added. The third input z, represents the carry from the previous lower significant position. Two outputs are necessary because the arithmetic sum of three binary digits ranges in value from 0 to 3 and binary 2 or 3 needs two digits. The two outputs are designated by the symbols S for sum and C for carry. The binary variable S gives the value of the least significant bit of the sum. The binary variable C gives the output of the carry. The name of the former stems from the fact that two half adders are needed to implement a full adder.

C=xy+xz+yz

=xy+(x’y+xy’)z

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