1. Basic Gates 3.1 Basic Digital Logic: NAND and NOR Gates ©Paul Godin Created September 2007 Last Update Sept 2009

2. Basic Gates 3.2 The Universal Gates: NAND and NOR

3. Basic Gates 3.3 Combinational logic ◊How would your describe the output of this combinational logic circuit? A B Z

4. Basic Gates 3.4 NAND Gate ◊The NAND gate is the combination of an NOT gate with an AND gate. The Bubble in front of the gate is an inverter. A B Z

5. Basic Gates 3.5 NAND Gate ◊IEEE Symbol ◊Boolean equations for the NAND gate: & A●B = x AB = x The triangle is the same as the bubble.

6. Basic Gates 3.6 Combinational logic ◊How would your describe the output of this combinational logic circuit? B A Z

7. Basic Gates 3.7 NOR gate ◊The NOR gate is the combination of the NOT gate with the OR gate. The Bubble in front of the gate is an inverter. B A Z

8. Basic Gates 3.8 NOR Gate ◊IEEE Symbol for a NOR gate: ◊Boolean equation for a NOR gate: ≥1 A+B = x

9. Basic Gates 3.9 Exercise 1 Complete the Truth Table for the NAND and NOR Gates InputOutput 00 01 10 11 InputOutput 00 01 10 11 NAND NOR Hint: Think of the AND and OR truth tables. The outputs for the NAND and NOR are inverted.

10. Basic Gates 3.10 Exercise 2 ◊Turn the NAND and NOR gates into inverter (NOT) gates. Hint: Look at the Truth Table.

11. Basic Gates 3.11 Exercise 3 Complete the following timing diagram A B Z A B Z

12. Basic Gates 3.12 Exercise 4 Complete the following timing diagram A B Z A B Z

13. Basic Gates 3.13 Exercise 5 ◊Draw the following circuit: AB + BC

14. Basic Gates 3.14 Exercise 6 ◊Describe the function of a NAND gate, starting with the term “If any input...” ◊Describe the function of a NOR gate, starting with the term “If any input...”

15. Basic Gates 3.15 DeMorgan

16. Basic Gates 3.16 DeMorgan Theorem ◊The DeMorgan Theorem describes a method for converting between AND/NAND and OR/NOR operations. ◊The theorem states: A ● B = A + B A + B = A ● B “Break the bar and change the sign”

17. Basic Gates 3.17 DeMorgan An example of DeMorgan: AB + BC ABC 1 2 3 4 Original Equation DeMorgan applied to NOR expression Double inversions cancel Simplified expression AB ● BC

18. Basic Gates 3.18 DeMorgan Exercise 1 Use DeMorgan to simplify the following expressions A+B+C AB AB + C+D “Break the bar and change the sign”

19. Basic Gates 3.19 Universal Gates ◊The NAND and NOR gates are considered Universal Gates. They can be used to create any other gate. ◊Using universal gates is an important aspect of digital logic design. Examples provided in class.

20. Basic Gates 3.20 NAND and NOR as NOT Z A Vcc A Z Method 1 AZ A Z Method 2 A Z

21. Basic Gates 3.21 NAND and NOR as AND Z B A Z B A AB = AB = A+B (DeMorgan) A Z B

22. Basic Gates 3.22 NAND and NOR as OR A+B = A+B = AB (DeMorgan) Z B A A Z B Z B A

23. Basic Gates 3.23 Universal Gates Exercise 1 ◊Convert NAND as NOR ◊Convert NOR as NAND

24. Basic Gates 3.24 Example: Universal Gates Convert the following circuit to NAND only: Convert each of the gates in the circuit to its NAND equivalent and progressively re-draw the circuit. Additional Examples given in class

25. Basic Gates 3.25 Active Input States

26. Basic Gates 3.26 Digital 1’s and 0’s ◊In a binary system, the logic 1’s are as important as the logic 0’s. A “0” is a signal also. ◊When the “0” forces a change it is called Active Low (the low causes the action). ◊When the “1” forces a change it is called Active High (the high causes the action).

27. Basic Gates 3.27 Comparison of Active States A logic 0 causes the LED to light up A logic 1 causes the LED to light up Vcc

28. Basic Gates 3.28 Comparison of Active Inputs A Z B Active Low Inputs Active High Output Active High Inputs Active Low Output Z B A

29. Alternate Gate Representations Basic Gates 3.29

30. Comparison of Active Inputs A Z B Active Low Inputs Active High Output Active High Inputs Active Low Output Z B A Basic Gates 3.30

31. Bubble to Bubble ◊Alternate gate representations can make circuit analysis faster. ◊A bubble attached to a bubble means the bubbles cancel. = Bubbles Cancel Basic Gates 3.31

32. Bubble to Bubble Cancellation Example 2 Z 1 A B C In this example with bubble-to-bubble representation, the output bubble from gate 1 cancels with the input bubble from gate 2. This makes it easy to quickly determine that if either A or B inputs are low, outcome Z is low. 2 Z 1 A B C Basic Gates 3.32

33. Alternate Representations ◊The “bubble” on a gate represents inversion. ◊In many cases it is easier to follow the circuit logic if “bubble” outputs are linked to “bubble” inputs ◊Cancelled bubbles helps make the active input state more easily visible for troubleshooting Basic Gates 3.33

34. Example use of Alternate Representation ◊The output is active when the input state is a 101 ◊Note how much easier it is to see the active input at a glance using a bubble instead of a NOT gate. Z C B A Basic Gates 3.34

35. Basic Gates 3.35 Example of Alternate Representation A Z B Z B A A Z B Equals A+B = AB (DeMorgan)

36. Basic Gates 3.36 Alternate Representation Z B A Z A Z A Z A Z B A B Z B A Z B A Z A Z B A B Z B A

37. Basic Gates 3.37 Alternate Representation Exercise Z A Z B A B Z A Z B A B Identify the gate that is alternately represented

38. Basic Gates 3.38 END 101101101 ©Paul R. Godin prgodin ° @ gmail.com