Lecture Notes: Creating XOR and XNOR Gates using NAND Gates

Introduction

• Speaker: Addressed to a viewer named Hak Mundal.
• Topic: Constructing XOR and XNOR gates using NAND gates.
• Objective: Efficiently design XOR and XNOR gates with minimal use of NAND gates.

XOR Gate Design

Standard Approach

• Formula: [ Y = AB' + A'B ]
• Implementation using Gates:
  • Requires separate gates for operations:
    • One gate for each ( AB' ) and ( A'B ).
    • Two gates for AND operation for each term.
    • Additional three gates for the OR operation.
  • Total Gates Required: 9 gates.

Efficient Approach

• Formula Conversion:

  • Start with ( Y = AB' + A'B ).
  • Modify to: ( AB' + A'A ) (since ( A'A = 0 ))
  • Further expand to: ( A' + B \times A + B \times A' + B' ).

• Simplification Using Double Complement:

  • Convert ( A' + B' ) and ( B + A' ) using a double complement.
  • Use De Morgan’s theorem to transform expressions into NAND logic:
    • Convert operations to ( (A'B)'' ) and ( (B'A)'' ).
  • Implementation:
    • Use one NAND gate for each intermediate operation.
    • Final result uses 4 NAND gates in total.

Process Recap

• Generate signal using the inputs A and B.
• Apply NAND between A and the generated signal.
• Apply NAND between B and the same signal.
• Combine results using another NAND gate to get ( A \oplus B ) (XOR).

XNOR Gate Design

• Implementation: Requires just one additional NAND gate compared to XOR.
• Design: ( A \odot B ).

Conclusion

• Efficient XOR gate design using only 4 NAND gates.
• Adding one more gate enables the XOR gate to function as an XNOR gate.
• Call to Action: Viewers are encouraged to comment with requests for other gate designs.