Binary Numbers and Logic Gates Lecture Notes

Introduction to Binary Numbers

• The term binary is associated with the number 2.
• Binary numbers consist of two possibilities: 0 or 1.
• In circuits:
  • 0 = Circuit off = false
  • 1 = Circuit on = true
  • Voltage representation: 0 volts (off) and typically 5 volts (on).

Buffer Gate

• Symbol: Triangle pointing right.
• Functionality:
  • Input = A, Output = A.
  • Truth Table:
    • Input (1) → Output (1)
    • Input (0) → Output (0)
• Circuit Representation:
  • Uses an NPN transistor.
  • Connect to a power source and a light-emitting diode (LED) representing output.
  • If voltage is applied to Input A (1), LED is on.
  • If no voltage (0), LED is off.

NOT Gate

• Symbol: Similar to Buffer but with a circle in front.
• Functionality:
  • Input = A, Output = A‘ (complement).
  • Truth Table:
    • Input (0) → Output (1)
    • Input (1) → Output (0)
• Circuit Representation:
  • Similar to Buffer but LED is connected across the collector and emitter of the transistor.
  • If Input A is on (1), LED is off (0).
  • If Input A is off (0), LED is on (1).

AND Gate

• Symbol: Standard AND gate symbol.
• Functionality:
  • Inputs: A and B; Output = A * B.
  • Truth Table:
    • Input (0, 0) → Output (0)
    • Input (1, 0) → Output (0)
    • Input (0, 1) → Output (0)
    • Input (1, 1) → Output (1)
• Circuit Representation:
  • Uses two NPN transistors in series.
  • LED is on only when both A and B are on (1).

OR Gate

• Symbol: Standard OR gate symbol.
• Functionality:
  • Inputs: A and B; Output = A + B.
  • Truth Table:
    • Input (0, 0) → Output (0)
    • Input (1, 0) → Output (1)
    • Input (0, 1) → Output (1)
    • Input (1, 1) → Output (1)
• Circuit Representation:
  • Uses two NPN transistors in parallel.
  • LED is on if either A or B is on.

NAND Gate

• Symbol: Similar to AND but with a circle at the output.
• Functionality:
  • Output = (A * B)' (complement of AND gate).
  • Truth Table:
    • Only output (0) when both A and B are 1.
• Can be constructed using an AND gate followed by a NOT gate.
• If both inputs A are connected, it acts as a NOT gate.

NOR Gate

• Symbol: Similar to OR but with a circle at the output.
• Functionality:
  • Output = (A + B)' (complement of OR gate).
  • Truth Table:
    • Output is 1 only when both A and B are 0.

Writing Functions from Block Diagrams

• Identify gate types and inputs.
• Use multiplication for AND gates and addition for OR gates.
• Example Process:
  1. For AND gate: Output = A * B.
  2. For OR gate: Output = A + B.

SOP and POS Expressions

• SOP (Sum of Products): Sum of multiple product terms.
• POS (Product of Sums): Product of multiple sum terms.
• Literals: Individual variables (complemented or uncomplemented).

Basic Rules of Boolean Algebra

1. Commutative Property: A + B = B + A; A * B = B * A.
2. Associative Property: (A + B) + C = A + (B + C); (A * B) * C = A * (B * C).
3. Identity Rule: A + 0 = A; A * 1 = A.
4. Null Property: A + 1 = 1; A * 0 = 0.
5. Complement Property: A + A' = 1; A * A' = 0.

Conclusion

• Understanding binary numbers, logic gates, and Boolean algebra is crucial for circuit design and logic functions.