Combinational Logic Circuits

A combinational circuit can be described by \(m\) Boolean functions, one for each output variable. Each output function is expressed in terms of the \(n\) input variables.

For \(n\) input variables, there are \(2^n\) possible combinations of the binary inputs. For each possible input combination, there is one possible value for each output variable. Thus, a combinational circuit can be specified with a truth table that lists the output values for each combination of input variables.

There are several combinational circuits that are employed extensively in the design of digital systems. These circuits are available in integrated circuits and are classified as standard components, such as adders, subtractors, comparators, decoders, encoders, and multiplexers.

Analysis of Combinational Circuits

Analysis of a combinational circuit determines its functionality, that is, the logic function that the circuit implements. This task starts with a given logic diagram and culminates with a set of Boolean functions, a truth table, or, possibly, an explanation of the circuit operation.

The first step in the analysis of a circuit is to make sure that it is combinational and not sequential.

The logic diagram of a combinational circuit has logic gates with no feedback paths or memory elements.

A feedback path is a connection from the output of one gate to the input of a second gate whose output forms part of the input to the first gate. Feedback paths in a digital circuit define a sequential circuit and must be analyzed by special methods.

To obtain the output Boolean functions of a combinational circuit from its logic diagram, the process involves:

With arbitrary, but meaningful, symbols, label the outputs of all gates whose inputs include at least one input of the circuit. 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. Find the Boolean functions for these gates.
Repeat the process outlined 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.

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

Determine the number of input variables in the circuit. For \(n\) inputs, form the \(2^n\) possible input combinations and list the binary numbers from \(0\) to \(\left(2^n - 1\right)\) in a table.
Label the outputs of selected gates with arbitrary symbols.
Obtain the truth table for the outputs of those gates whose set of inputs consists of only inputs to the circuit.
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 involves the following steps:

From the specifications of the circuit, determine the required number of inputs and outputs and assign a symbol to each.
Derive the truth table that defines the required relationship between inputs and outputs.
Obtain the simplified Boolean functions for each output as a function of the input variables.
Draw the logic diagram and verify the correctness of the design (manually or by simulation).

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Sequential Logic Circuits

A change in state of the flip-flops is initiated only by a clock pulse transition. When a clock pulse is not active, the feedback loop between the value stored in the flip-flop and the value formed at the input to the flip-flop is effectively broken because the flip-flop outputs cannot change even if the outputs of the combinational circuit driving their inputs change. Thus, the transition from one state to the next occurs only at predetermined intervals dictated by the clock pulses.

A sequential circuit consists of a combinational circuit to which memory (storage) elements are connected to form a feedback path. The storage elements are devices capable of storing binary information. The binary information stored in these elements at any given time defines the state of the sequential circuit at that time.

If the clock (synchronizing) pulses arrive at a regular interval, the combinational logic must respond to a change in the state of the flip-flop in time to be updated before the next pulse arrives. Propagation delays play an important role in determining the minimum interval between the clock pulses that will allow the circuit to operate correctly.

Thus, a sequential circuit is specified by a time sequence of inputs, outputs, and internal states. In contrast, the outputs of combinational logic depend on only the present values of the inputs.