1 / 41

Combinatorial Circuits: Boolean Algebra and Circuit Design

Learn how to design electronic circuits using Boolean algebra and explore the application of logical gates in switching circuits. Discover how to implement various gates and minimize Boolean expressions using Karnaugh maps.

pilkington
Download Presentation

Combinatorial Circuits: Boolean Algebra and Circuit Design

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 10.3 and 10.4: Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

  2. Learning Objectives • Explore the application of Boolean algebra in the design of electronic circuits • Learn the application of Boolean algebra in switching circuits Discrete Mathematical Structures: Theory and Applications

  3. Logical Gates and Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

  4. Logical Gates and Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

  5. Logical Gates and Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

  6. Logical Gates and Combinatorial Circuits • In circuitry theory, NOT, AND, and OR gates are the basic gates. Any circuit can be designed using these gates. The circuits designed depend only on the inputs, not on the output. In other words, these circuits have no memory. Also these circuits are called combinatorial circuits. • The symbols NOT gate, AND gate, and OR gate are also considered as basic circuit symbols, which are used to build general circuits. The word circuit instead of symbol is also used. Discrete Mathematical Structures: Theory and Applications

  7. Logical Gates and Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

  8. Discrete Mathematical Structures: Theory and Applications

  9. Discrete Mathematical Structures: Theory and Applications

  10. Discrete Mathematical Structures: Theory and Applications

  11. Discrete Mathematical Structures: Theory and Applications

  12. Discrete Mathematical Structures: Theory and Applications

  13. Discrete Mathematical Structures: Theory and Applications

  14. Discrete Mathematical Structures: Theory and Applications

  15. Discrete Mathematical Structures: Theory and Applications

  16. Discrete Mathematical Structures: Theory and Applications

  17. Discrete Mathematical Structures: Theory and Applications

  18. Discrete Mathematical Structures: Theory and Applications

  19. Discrete Mathematical Structures: Theory and Applications

  20. Examples 2 and 3, p. 714 Discrete Mathematical Structures: Theory and Applications

  21. Discrete Mathematical Structures: Theory and Applications

  22. Discrete Mathematical Structures: Theory and Applications

  23. Discrete Mathematical Structures: Theory and Applications

  24. Discrete Mathematical Structures: Theory and Applications

  25. Discrete Mathematical Structures: Theory and Applications

  26. Discrete Mathematical Structures: Theory and Applications

  27. Discrete Mathematical Structures: Theory and Applications

  28. Discrete Mathematical Structures: Theory and Applications

  29. Logical Gates and Combinatorial Circuits • The diagram in Figure 12.32 represents a circuit with more than one output. Discrete Mathematical Structures: Theory and Applications

  30. A half adder is a circuit that accepts as input two binary digitsx and y, and produces as output the sum bit s and the carry bit c. Discrete Mathematical Structures: Theory and Applications

  31. Discrete Mathematical Structures: Theory and Applications

  32. Discrete Mathematical Structures: Theory and Applications

  33. Logical Gates and Combinatorial Circuits • A NOT gate can be implemented using a NAND gate (see Figure 12.36(a)). • An AND gate can be implemented using NAND gates (see Figure 12.36(b)). • An OR gate can be implemented using NAND gates (see Figure12.36(c)). Discrete Mathematical Structures: Theory and Applications

  34. Logical Gates and Combinatorial Circuits • Any circuit which is designed by using NOT, AND, and OR gates can also be designed using only NAND gates. • Any circuit which is designed by using NOT, AND, and OR gates can also be designed using only NOR gates. Discrete Mathematical Structures: Theory and Applications

  35. Discrete Mathematical Structures: Theory and Applications

  36. Discrete Mathematical Structures: Theory and Applications

  37. Logical Gates and Combinatorial Circuits • The Karnaugh map, or K-map for short, can be used to minimize a sum-of-product Boolean expression. Discrete Mathematical Structures: Theory and Applications

  38. Discrete Mathematical Structures: Theory and Applications

  39. Discrete Mathematical Structures: Theory and Applications

  40. Logical Gates and Combinatorial Circuits • First mark the 1s that cannot be paired with any other 1. Put a circle around them. • Next, from the remaining 1s, find the 1s that can be combined into two square blocks, i.e., 1 x 2 or 2 x 1 blocks, and in only one way. • Next, from the remaining 1s, find the 1s that can be combined into four square blocks, i.e., 2 x 2, 1 x 4, or 4 x 1 blocks, and in only one way. • Next, from the remaining 1s, find the 1s that can be combined into eight square blocks, i.e., 2 x 4 or 4 x 2 blocks, and in only one way. • Next, from the remaining 1s, find the 1s that can be combined into 16 square blocks, i.e., a 4 x 4 block. (Note that this could happen only for Boolean expressions involving four variables.) • Finally, look at the remaining 1s, i.e., the 1s that have not been grouped with any other 1. Find the largest blocks that include them. Discrete Mathematical Structures: Theory and Applications

  41. Discrete Mathematical Structures: Theory and Applications

More Related