Basic Digital Logic: NAND and NOR Gates Technician Series ©Paul Godin Last Update Dec 2014

The Universal Gates: NAND and NOR

Combinational logic ◊How would your describe the output of this combinational logic circuit? A B Z Basic Gates 2.3

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 Basic Gates 2.4

NAND Gate ◊IEEE Symbol ◊Boolean equations for the NAND gate: & A●B = x AB = x The triangle is the same as the bubble. Basic Gates 2.5

Combinational logic ◊How would your describe the output of this combinational logic circuit? B A Z Basic Gates 2.6

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 Basic Gates 2.7

NOR Gate ◊IEEE Symbol for a NOR gate: ◊Boolean equation for a NOR gate: ≥1 A+B = x Basic Gates 2.8

Exercise 1 Complete the Truth Table for the NAND and NOR Gates InputOutput InputOutput NAND NOR Hint: Think of the AND and OR truth tables. The outputs for the NAND and NOR are inverted. Basic Gates 2.9

Exercise 2 ◊Turn the NAND and NOR gates into inverter (NOT) gates. Hint: Look at the Truth Table. Basic Gates 2.10

Exercise 3 ◊Draw the following circuit: AB + BC Basic Gates 2.11

Exercise 4 ◊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...” Basic Gates 2.12

DeMorgan

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” Basic Gates 2.14

DeMorgan An example of DeMorgan: AB + BC ABC Original Equation DeMorgan applied to NOR expression Double inversions cancel Simplified expression AB ● BC Basic Gates 2.15

DeMorgan Exercise 1 Use DeMorgan to simplify the following expressions A+B+C AB AB + C+D “Break the bar and change the sign” Basic Gates 2.16

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. Basic Gates 2.17

NAND and NOR as NOT Z A Vcc A Z Method 1 AZ A Z Method 2 A Z Basic Gates 2.18

NAND and NOR as AND Z B A Z B A AB = AB = A+B (DeMorgan) A Z B Basic Gates 2.19

NAND and NOR as OR A+B = A+B = AB (DeMorgan) Z B A A Z B Z B A Basic Gates 2.20

Universal Gates Exercise 1 ◊Convert NAND as NOR ◊Convert NOR as NAND Basic Gates 2.21

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 Basic Gates 2.22

Active Input States

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). Basic Gates 2.24

Active Low State ◊Many, many devices use the logic low to cause something to change. ◊Resets and presets on digital devices ◊Communication systems ◊Active low inputs are indicated with either an overbar on the input label, and/or a bubble in front of the output. Basic Gates 2.25 Reset

Comparison of Active States A logic 0 causes the LED to light up A logic 1 causes the LED to light up Vcc Basic Gates 2.26

Alternate Gate Representations

Comparison of Active Inputs A Z B Active Low Inputs Active High Output Active High Inputs Active Low Output Z B A Basic Gates 2.28

Bubble to Bubble ◊Alternate gate representations can make circuit analysis faster. ◊A bubble attached to a bubble means the bubbles cancel. Basic Gates 2.29 = Bubbles Cancel

Bubble to Bubble Cancellation Example Basic Gates 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

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 2.31

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 2.32

Example of Alternate Representation A Z B Z B A A Z B Equals A+B = AB (DeMorgan) Basic Gates 2.33

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 Basic Gates 2.34

Alternate Representation Exercise Z A Z B A B Z A Z B A B Identify the gate that is alternately represented Basic Gates 2.35

Alternate Representation Exercise ANSWERS Z A Z B A B Z A Z B A B Identify the gate that is alternately represented Basic Gates 2.36

END ©Paul R. Godin prgodin gmail.com Basic Gates 2.37