Application to Switching Circuits (Contd...)

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Logic Gates

The basic building blocks for a complex digital system are known as logic gates.

A gate is a simple electronic circuit or device that performs logical functions. It has one or more inputs and one output gate and are also called binary logic gates, as 1 and 0 are the inputs and outputs.

We shall discuss the three basic gates known as 'AND', 'OR' and 'NOT' gates whose input - output tables are similar to conjunction, disjunction and negation respectively with T = 1 and F = 0.

AND Gate

An 'AND' gate is a Boolean function with two input variables x₁, x₂, defined by

f(x₁, x₂) = x₁, x₂

where x₁, x₂ {0, 1}

The input-output table for 'AND' gate is given below, which is similar to Table 3 (A circuit with two switches connected in series)

Table 10

The following figure represents 'AND' gate

OR Gate

An 'OR' gate is a Boolean function with two variables defined by

f(x₁, x₂) = x₁ + x₂ x₁, x₂ {0, 1}

The above figure shows the 'OR' gate and the input-output table for OR gate is given in Table 11 which is similar to Table 6 (A circuit with two switches connected in parallel)

Table 11

NOT Gate

A NOT gate is the Boolean function defined by

f(x) = x' x{0, 1}

NOT gate is shown in figure and the input/output table is given in Table 12.

Table 12

Example:

Find the output for the given input.

x₁ = 1, x₂ = 1 from the circuit given in figure.

The Boolean function for the given circuit is

S = f(x₁, x₂) = (x₁ . x₂)' + x₁

If x₁ = 1, x₂ = 1

S = (x₁ . x₂)' + x₁ = (1 . 1)' + 1

= (1)' + 1 = 0 + 1

= 1

Combinatorial Circuit

A circuit in which the output is uniquely defined for each combination of inputs x₁, x₂, x₃ is called a combinatorial circuit.

Equivalent Circuit

Two combinatorial circuits, each having inputs x₁, x₂, x₃, x₄,……x_n and a single output are said to be equivalent if their input/output tables are identical.

Example:

Simplify the combinatorial circuit for the given figure.

The given circuit is x₁.x₂ + x₁'x₂ + x₁x₂'

= x₂(x₁ + x₁') + x₁x₂' Axiom 4(a)

= x₂ . 1 + x₁x₂' Axiom 5(a)

= x₂ + x₁x₂' Axiom 2(b)

= x₁x₂' + x₂ Axiom 3(a)

The simplified combinatorial circuit is