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| Application to Switching Circuits (Contd...) |
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| The basic building blocks for a complex digital system are known as logic gates. |
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| 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. |
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| 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. |
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| An 'AND' gate is a Boolean function with two input variables x1, x2, defined by |
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| f(x1, x2) = x1, x2 |
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where x1, x2  {0, 1} |
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| The input-output table for 'AND' gate is given below, which is similar to Table 3 (A circuit with two switches connected in series) |
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| Table 10 |
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| The following figure represents 'AND' gate |
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| An 'OR' gate is a Boolean function with two variables defined by |
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f(x1, x2) = x1 + x2 x1, x2  {0, 1} |
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| 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) |
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| Table 11 |
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| A NOT gate is the Boolean function defined by |
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f(x) = x' x {0, 1} |
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| NOT gate is shown in figure and the input/output table is given in Table 12. |
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| Table 12 |
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| Example: |
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| Find the output for the given input. |
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| x1 = 1, x2 = 1 from the circuit given in figure. |
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| The Boolean function for the given circuit is |
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| S = f(x1, x2) = (x1 . x2)' + x1 |
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| If x1 = 1, x2 = 1 |
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| S = (x1 . x2)' + x1 = (1 . 1)' + 1 |
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| = (1)' + 1 = 0 + 1 |
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| = 1 |
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| A circuit in which the output is uniquely defined for each combination of inputs x1, x2, x3 is called a combinatorial circuit. |
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| Two combinatorial circuits, each having inputs x1, x2, x3, x4,……xn and a single output are said to be equivalent if their input/output tables are identical. |
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| Example: |
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| Simplify the combinatorial circuit for the given figure. |
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| The given circuit is x1.x2 + x1'x2 + x1x2' |
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| = x2(x1 + x1') + x1x2' Axiom 4(a) |
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| = x2 . 1 + x1x2' Axiom 5(a) |
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| = x2 + x1x2' Axiom 2(b) |
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| = x1x2' + x2 Axiom 3(a) |
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| The simplified combinatorial circuit is |
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