Essay: Combinational Circuits / Sequential Circuits

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  • Combinational Circuits / Sequential Circuits
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Introduction:

 

Combinational Circuits

 

Sequential Circuits

 

The output depends only on the input present at that input.

 

The output depends not only on the inputs but also on the past outputs.

 

There is no feedback.

 

Output is fedback to input.

 

Memory unit is not required.

 

Memory unit is required.

 

Eg: MUX, Demux, Encoder, Decoder.

 

Eg: Shift registers, counters.

 

2.1 Combinational logic

 

With combinational logic, the circuit produces the same output regardless of the order the inputs are changed. There are circuits which depend on the when the inputs change, these circuits are called sequential logic.

 

Practical circuits will have a mix of combinational and sequential logic, with sequential logic making sure everything happens in order and combinational logic performing functions like arithmetic, logic or conversion.

 

Design using gates

 

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 is 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 not only on the present but also the past state of inputs.

 

 

 

Generalized Combinational Circuit

 

The figure shows the block schematic representation of a generalized combinational circuit having n input variables and m output variables or simply outputs.

 

Since the number of input variables is n, there are 2n possible combinations of bits at the input.

 

Each output can be expressed in terms of input variables by a Boolean expression with the result that the generalized system of above figure can be expressed by m Boolean expressions.

 

As an illustration, Boolean expressions describing the function of a four-input OR/NOR gate are given as,

 

 

 

Problem

 

Let us assume that the input frequency of a 7497 binary rate multiplier is 64 KHz. Let us derive its output if the multiplier word is 1011.

 

Solution

2.2 Representation of logic functions

 

Three Representations of Logic Functions

 

1. Logical expressionANDORNOT

 

2. Truth Table X.Y X+Y X =X’

 

AND Operation

 

AND operation is represented by C = A • B. Its associated TRUTH TABLE is shown below.

 

A truth table gives the value of output variable (here C) for all combinations of input variable values (here A and B). Thus in an AND operation, the output will be 1 (True) only if all of the inputs are 1 (True).

 

 

 

OR operation

 

OR operation is represented by C = A + B. Here A, B & C are logical (Boolean) variables and the + sign represents the logical addition called an ‘OR’ operation.

 

Its associated TRUTH TABLE is shown below. Thus in an OR operation, the output will be 1 (True) if either of the inputs is 1 (True).

 

If both inputs are 0 (False), only then the output will be 0 (False). Notice that though the symbol + is used, the logical addition described above does not follow the rules of normal arithmetic addition.

 

 

 

NOT operation

 

NOT operation is represented by,

 

The NOT gate has only one input which is then inverted by the gate. Here is the ‘complement’ of A.

 

The truth table for the operation are shown below:

 

 

 

Circuit Diagram / Schematic

 

The symbol for the operation (called an AND gate) is shown in figure.

 

 

 

The symbol for the operation (called an OR gate) is shown in figure.

 

 

 

The symbol for the operation (called NOT gate) is shown in figure.

 

 

 

2.3 SOP and POS forms

 

 

Sum-of-Products Implementation

A Sum-Of-Products Boolean expression is literally a set of Boolean terms added (summed) together, each term being a multiplicative (product) combination of Boolean variables.

sum-of-products-expression = term1 + term2 … + term n

Product terms that include all of the input variables (or their inverses) are called minterms.

In a sum-of-products expression, we form a product of all the input variables (or their inverses) for each row of the truth table for which the result is logic 1. The output is the logical “sum” of these minterms.

Sum-Of-Products expressions are easy to generate from truth tables as shown in the example below by determining which rows of the table have an output of 1, writing one product term for each rows and finally summing all the product terms. This creates a Boolean expression representing the truth table as a whole.

Sum-Of-Products expressions lend themselves well to implementation as a set of AND gates (products) feeding into a single OR gate (sum).

Truth table

 

The output can be expressed as,

 

Product-of-Sums Implementation

 

An alternative to generating a Sum-Of-Products expression to account for all the “high” (1) output conditions in the truth table is to generate a Product-Of-Sums or POS expression, to account for all the “low” (0) output conditions instead.

 

POS Boolean expressions can be generated from truth tables quite easily by determining which rows of the table have an output of 0, writing one sum term for each row and finally multiplying all the sum terms.

 

This creates a Boolean expression representing the truth table as a whole. These “sum” terms that include all of the input variables (or their inverses) are called maxterms.

 

For POS implementation, the output variable is the logical product of maxterms. Product-Of-Sums expressions lend themselves well to implementation as a set of OR gates (sums) feeding into a single AND gate (product).

 

Truth table

 

 

 

The output can be expressed as,

 

 

 

Minterm and Maxterm:

 

A product term containing all the variables of the function in either complemented or un complemented form is called a minterm.

 

A sum term containing all the variables of the function in either complemented or un complemented form is called a maxterm.

 

Problem

 

For the given circuit, let us derive an algebraic expression in sop form.

 

Solution

 

 

 

First we minimize and implement the following multiple output functions in SOP form.

 

f1 = Σm (0,2,6,10,11,12,13) + d (3,4,5,14,15)

 

f2 = Σm (1,2,6,7,8,13,14,15) + d (3,5,12)

 

Solution

 

 

 

The K maps are filled ones and don’t cares.

 

After reduction we find that CD occurs both in f1 and f2. So it can be shared.

 

 

 

Logic Circuit Implementation

 

Here we can see an example for simplifying and implementing the following SOP function using NOR gates. f(A, B, C, D) = Σm (0,1,4, 5,10,11,14,15).

 

Solution

 

 

 

Lets see an another example to show that a function expressed as a sum of its minterms is equivalent to a function expressed as a product of its maximum terms.

 

Solution

 

Let us assume that a function .

 

SOP:

 

 

 

POS:

 

 

 

Lets see an example for expressing the function in both Canonical SOP form and Canonical POS form.

 

Solution

 

(a)Sum of Product (SOP)

 

 

 

Y = m7 + m6 + m5 + m3 + m5 + m1

 

Y = Σ(1, 4, 5, 6, 7).

 

(b)Product of Sum (POS)

 

 

 

Y = M2 M3 M0 = M0 M2 M3

 

Y = π(0, 2, 3).

 

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