1 / 19

Understanding Digital Logic Circuits: Basics, Gates, and Boolean Expressions

This chapter explores the foundation of digital logic circuits, detailing their resemblance to logic tables and the types of switches and circuits involved. It discusses combinational circuits, which derive outputs based on current inputs, and the rules governing their construction. Boolean variables and expressions are introduced, illuminating how they translate into logic circuits using basic gates. The significance of NAND and NOR gates for circuit simplification is also covered, providing a comprehensive overview of creating and analyzing digital logic circuits.

una
Download Presentation

Understanding Digital Logic Circuits: Basics, Gates, and Boolean Expressions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 1 The Logic of Compound Statements

  2. Section 1.4 Digital Logic Circuits

  3. Digital Circuits • Electrical circuits can be fashioned to mimic logic tables. • Types of switches: • open • closed • Types of circuits: • series • parallel

  4. Switching Table • Switches in series • closed/on => T • open/off => F

  5. Switching Table • Switches in parallel • closed/on => T • open/off => F

  6. Basic Digital Logic Gates

  7. Combinational Circuits • Combinational circuits are composed of one or more basic gates where the output of the circuit is based on the input at that instant in time. • Rules of Combinational Circuits • Never combine two input wires. • A single input wire can be split and used as input for two separate gates. • An output wire can be used as input. • No output of a gate can feedback into that gate. • Sequential circuits are circuits that include feedback. Their output depends on previous input. These circuits are used to build circuits that can remember (memory circuits).

  8. Example

  9. Input-Output Table • Input-output table is a truth table for a combinational circuit. It shows the output of the circuit given a set of inputs.

  10. Example P v Q (P v Q) ^ ~(P ^ Q) ~(P ^ Q) P ^ Q

  11. Boolean • A combinational circuit can be expressed as a Boolean expression. • George Boolean was an English mathematician who founded symbolic logic. • Boolean variable is a variable that has only two possible values (T/F, on/off, 1/0). • Boolean expression is composed of Boolean variables and connectives (~, v, ^ )

  12. Boolean Expression Circuits • A Boolean expression can be converted to a combinational digital logic circuit by using the Boolean variables as inputs and matching the connectives (~, v, ^) with their gate equivalent (NOT, OR, AND). • Example • (~P ^ Q) v ~Q

  13. Circuit from I/O Table • A circuit can be constructed from any I/O table. • A circuit constructed in this form will be composed of a set of AND gates connected by OR gates. R^S v ~R^S v R^~S

  14. Example 1^1^1 v 1^0^1 v 1^0^0 P^Q^Rv P^~Q^Rv P^~Q^~R

  15. Equivalent Circuits • Two circuits are equivalent if there I/O tables are equivalent. • As with logic expressions, digital circuits may be simplified through logic theorem 1.1.1, aka Boolean Algebra.

  16. Example • ((P ^ ~Q) V (P ^ Q)) ^ Q • (P ^ (~Q V Q)) ^ Q (distributive) • (P ^ (Q v ~Q)) ^ Q (commutative) • (P ^ t) ^ Q (negation) • P ^ Q (identity) • Inspection of the I/O table reveals the simplified circuit.

  17. NAND and NOR Gates • NAND or NOR gates can be used to simplify a circuit as they are primitive gates, i.e. all gates can be built from them. (NOT, AND, OR, XOR, etc.)

  18. NAND and NOR • NAND • logic symbol is (Sheffer Stroke) | • P|Q  ~(P ^ Q) • NOR • logic symbol is (Peirce Arrow)  • PQ  ~(P v Q)

  19. NAND (Sheffer Stroke) Example • Show that the Sheffer Stroke (NAND) can be used to implement ~ (NOT) • ~P  P | P • ~P  ~(P ^ P) (idempotent) •  P | P (definition of |)

More Related