Combinational circuits

COMBINATIONAL LOGIC

PRE-REQUISITE DISCUSSION

The term combinational comes to us from mathematics. In mathematics a combination is an unordered set, which is a formal way to say that nobody cares which order the items came in.

Logic circuits for digital systems may be combinational or sequential. A combinational circuit consists of logic gates whose outputs at any time are determined from only the present combination of inputs. A combinational circuit performs an operation that can be specified logically by a set of Boolean functions. In contrast, sequential circuits employ storage elements in addition to logic gates. Their outputs are a function of the inputs and the state of the storage elements.

COMBINATIONAL CIRCUITS

A combinational circuit is one where the output at any time depends only on the present combination of inputs at that point of time with total disregard to the past state of the inputs. The logic gate is the most basic building block of combinational logic. The logical function performed by a combinational circuit is fully defined by a set of Boolean expressions. The other category of logic circuits, called sequential logic circuits, comprises both logic gates and memory elements such as flip-flops. Owing to the presence of memory elements, the output in a sequential circuit depends upon not only the present but also the past state of inputs.

A combinational circuit consists of input variables, logic gates, and output variables. Combinational logic gates react to the values of the signals at their inputs and produce the value of the output signal, transforming binary information from the given input data to a required output data.

The n-input binary variables come from an external source; the m- output variables are produced by the internal combinational logic circuit and go to an external destination.

ANALYSIS PROCEDURE

The analysis of a combinational circuit requires that we determine the function that the circuit implements. The analysis can be performed manually by finding the Boolean functions or truth table or by using a computer simulation program.

The first step in the analysis is to make that the given circuit is combinational or sequential. Once the logic diagram is verified to be combinational, one can proceed to obtain the output Boolean functions or the truth table.

To obtain the output Boolean functions from a logic diagram,

·  Label all gate outputs that are a function of input variables with arbitrary symbols or names. Determine the Boolean functions for each gate output.

·  Label the gates that are a function of input variables and previously labeled gates with other arbitrary symbols or names. Find the Boolean functions for these gates.

·    Repeat the process in step 2 until the outputs of the circuit are obtained.

·  By repeated substitution of previously defined functions, obtain the output Boolean functions in terms of input variables.

Logic diagram for analysis example

To obtain the truth table directly from the logic diagram without going through the derivations of Boolean functions,

· Determine the number of input variables in the circuit. For n-inputs, form the 2ⁿ possible input combinations and list the binary numbers from 0 to 2ⁿ-1 in a table.

·                    Label the outputs of selected gates with arbitrary symbols.

· Obtain the truth table for the outputs of those gates which are a function of the input variables only.

· Proceed to obtain the truth table for the outputs of those gates which are a function of previously defined values until the columns for all outputs are determined.

DESIGN PROCEDURE

The design of combinational circuits starts from the specification of the design objective and culminates in a logic circuit diagram or a set of Boolean functions from which the logic diagram can be obtained. The procedure involved involves the following steps,

ü From the specifications of the circuit, determine the required number of inputs and outputs and assign a symbol to each.

ü Derive the truth table that defines the required relationship between inputs and outputs.

ü Obtain the simplified Boolean functions for each output as a function of the input variables.

ü Draw the logic diagram and verify the correctness of the design.