Design NAND gate using NOR gate only.

Q.) Design NAND gate using NOR gate only.

 Subject: Digital Systems

To design a NAND gate using only NOR gates, you can utilize the De Morgan's theorem, which states that the negation of a conjunction (AND) of two propositions is logically equivalent to the disjunction (OR) of their negations, and vice versa.

Step 1: Understanding NAND and NOR Gates

  • A NAND (Not-AND) gate produces a FALSE output when all of its inputs are TRUE, and a TRUE output otherwise. It implements the logical operation (A \ NAND \ B = \overline{A \ AND \ B}).

  • A NOR (Not-OR) gate is the complement of the OR gate. It produces a FALSE output when all of its inputs are TRUE, and a TRUE output otherwise. It implements the logical operation (A \ NOR \ B = \overline{A \ OR \ B}).

Step 2: De Morgan's Theorem

De Morgan's theorem states that:

  • ( \overline{A \ AND \ B} = \overline{A} \ OR \ \overline{B})
  • ( \overline{A \ OR \ B} = \overline{A} \ AND \ \overline{B})

Step 3: Designing NAND using NOR Gates

To design a NAND gate using NOR gates, you can apply De Morgan's theorem to the NAND operation:

(A \ NAND \ B = \overline{A \ AND \ B})

Using De Morgan's theorem, we can rewrite this as:

(A \ NAND \ B = \overline{A} \ OR \ \overline{B})

This shows that the NAND operation can be implemented using two NOR gates.

Implementation

  1. Connect the two inputs (A) and (B) to the inputs of a NOR gate.
  2. Connect the outputs of this NOR gate (Y_1) to one of the inputs of another NOR gate.
  3. Connect the negated inputs (\overline{A}) and (\overline{B}) to the other inputs of the second NOR gate.
  4. The output (Y_2) from the second NOR gate is the NAND output, which will be (TRUE) only when both inputs (A) and (B) are (FALSE).

Example

Consider a NAND gate with inputs (A = 0) and (B = 1).

  1. The first NOR gate produces output (Y_1 = \overline{A \ OR \ B} = \overline{0 \ OR \ 1} = \overline{1} = 0).
  2. The second NOR gate produces output (Y_2 = \overline{Y_1 \ OR \ \overline{B}} = \overline{0 \ OR \ 0} = 1).

Therefore, the NAND gate output is (Y_2 = 1), which is correct according to the NAND truth table.

Conclusion

By utilizing De Morgan's theorem and connecting NOR gates in a specific configuration, we have designed a NAND gate using only NOR gates. This illustrates the concept of designing complex logic gates from simpler ones and demonstrates the versatility of NOR gates in implementing various logic functions.