Logic Gates Notes

Logic Gates

Chapter Goals

  • Identify the basic gates and describe the behavior of each.

  • Describe how gates are implemented using transistors.

  • Combine basic gates into circuits.

  • Describe the behavior of a gate or circuit using Boolean expressions, truth tables, and logic diagrams.

Computers and Electricity

  • Gate: A device that performs a basic operation on electrical signals.

  • Circuits: Gates combined to perform more complicated tasks.

Describing the Behavior of Gates and Circuits

  • Boolean Expressions: Utilizes Boolean algebra, a mathematical notation for expressing two-valued logic.

  • Logic Diagrams: Graphical representation of a circuit with symbols representing each gate.

  • Truth Tables: A table that shows all possible input values and the associated output values.

Types of Gates

  • Six main types of gates:

    • NOT

    • AND

    • OR

    • XOR (Exclusive OR)

    • NAND (NOT AND)

    • NOR (NOT OR)

  • Logic diagrams typically use black and white, with gates distinguished by shape.

Focus Gates for this Course

  • NOT, AND, and OR gates are the primary focus. The other gates are combinations of these.

NOT Gate

  • Accepts one input signal (0 or 1) and returns the opposite as output.

Various Representations of a NOT Gate

  • Standard symbol and representations may vary among textbooks or resources.

AND Gate

  • Accepts two input signals.

  • Outputs 1 only if both inputs are 1; otherwise, outputs 0.

Various Representations of an AND Gate

  • Similar to NOT, may vary in representations.

OR Gate

  • Accepts two input signals.

  • If both inputs are 0, output is 0; otherwise, output is 1.

Various Representations of an OR Gate

  • Visual variations in diagrams.

Review of Gate Processing

  • NOT Gate: Inverts its single input.

  • AND Gate: Produces 1 only if both input values are 1.

  • OR Gate: Produces 0 if both input values are 0.

Gates with More Inputs

  • Gates can be designed to accept three or more input values:

    • A three-input AND gate outputs 1 only if all inputs are 1.

Constructing Gates with Transistors

  • Transistor: Acts as a wire (conducts) or as a resistor (blocks) based on current signal voltage.

  • Functions like a switch, made from semiconductor material.

Circuits

  • Combinational Circuits: Output is determined exclusively by current inputs.

  • Sequential Circuits: Output depends on inputs and the existing state of the circuit.

Combining Gates into Circuits

  • Gates are combined such that the output of one gate can serve as the input for another.

Examples of Combinational Circuits

  • A circuit with three inputs requires 8 rows in its truth table.

  • Example Boolean expression for a circuit: (A B + A C).

  • Comparison with truth tables is crucial for understanding.

Circuit Equivalence

  • Equivalence: Two circuits that yield the identical output for the same input.

  • Boolean algebra facilitates circuit design through mathematical principles.

  • Example of distributive law: A(B + C) = AB + AC.

Properties of Boolean Algebra

  • Commutative Law: AB = BA (AND) / A + B = B + A (OR)

  • Associative Law: (AB)C = A(BC) (AND) / (A + B) + C = A + (B + C) (OR)

  • Distributive Law: A(B + C) = AB + AC / A + (BC) = (A + B)(A + C)

  • Identity: A1 = A / A0 = 0 / A + A' = 1

  • Complement: A(A') = 0 / A + 0 = A.

  • DeMorgan's Law: (AB)' = A' + B' / (A + B)' = A'B'.

Conclusion

  • Understanding logic gates and their behaviors lays the foundation for designing and analyzing digital circuits.