logic gates functions.pptx
- 1. Unit 2 Logic Gates and Functions
- 2. Unit 2: Logic Gates and Functions Overview • Introduction to Digital Logic Gates • Truth table • Symbol • Universal Gates (NAND/NOR) • XOR and XNOR Gates • Logic Chips • Logic Functions • Logical Equivalence • Standard Forms (SOP/POS)
- 3. Introduction to Digital Logic Basics Hardware consists of a few simple building blocks These are called logic gates AND, OR, NOT, … NAND, NOR, XOR, … Logic gates are built using transistors NOT gate can be implemented by a single transistor AND gate requires 3 transistors Transistors are the fundamental devices Pentium consists of 3 million transistors Compaq Alpha consists of 9 million transistors Now we can build chips with more than 100 million transistors
- 4. Basic Concepts Simple gates AND OR NOT Functionality can be expressed by a truth table A truth table lists output for each possible input combination Precedence NOT > AND > OR F = A B + A B = (A (B)) + ((A) B)
- 6. Basic Concepts (cont.) Additional useful gates NAND NOR XOR NAND = AND + NOT NOR = OR + NOT XOR implements exclusive-OR function NAND and NOR gates require only 2 transistors AND and OR need 3 transistors!
- 7. Basic Concepts (cont.) Proving NAND gate is universal
- 9. Basic Concepts (cont.) Proving NOR gate is universal
- 11. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved Floyd, Digital Fundamentals, 10th ed The XOR gate produces a HIGH output only when the inputs are at opposite logic levels. The truth table is The XOR Gate Inputs A B X Output 0 0 0 1 1 0 1 1 0 1 1 0 A B X A B X = 1 The XOR operation is written as X = AB + AB. Alternatively, it can be written with a circled plus sign between the variables as X = A + B. XOR and XNOR Gates
- 12. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved Floyd, Digital Fundamentals, 10th ed Example waveforms: A X Notice that the XOR gate will produce a HIGH only when exactly one input is HIGH. The XOR Gate B A B X A B X = 1
- 13. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved Floyd, Digital Fundamentals, 10th ed The XNOR gate produces a HIGH output only when the inputs are at the same logic level. The truth table is The XNOR Gate Inputs A B X Output 0 0 0 1 1 0 1 1 1 0 0 1 A B X A B X The XNOR operation can be shown as X = AB + AB. = 1
- 14. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved Floyd, Digital Fundamentals, 10th ed Example waveforms: A X Notice that the XNOR gate will produce a HIGH when both inputs are the same. This makes it useful for comparison functions. The XNOR Gate B A B X A B X = 1
- 15. Logic Chips
- 16. Logic Chips (cont.) Integration levels SSI (small scale integration) Introduced in late 1960s 1-10 gates (previous examples) MSI (medium scale integration) Introduced in late 1960s 10-100 gates LSI (large scale integration) Introduced in early 1970s 100-10,000 gates VLSI (very large scale integration) Introduced in late 1970s More than 10,000 gates
- 17. Logic Functions Number of functions With N logical variables, we can define 2 N combination of inputs A function relates outputs to inputs Some of them are useful AND, NAND, NOR, XOR, … Some are not useful: Output is always 1 Output is always 0
- 18. Logic Functions Logical functions can be expressed in several ways: Truth table Logical expressions Graphical form Example: Majority function Output is one whenever majority of inputs is 1 We use 3-input majority function
- 19. Logic Functions (cont.) 3-input majority function A B C F 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 Logical expression form F = A B + B C + A C
- 20. Logical Equivalence All three circuits implement F = A B function
- 21. Logical Equivalence Derivation of logical expression from a circuit Trace from the input to output Write down intermediate logical expressions along the path
- 22. Logical Equivalence (cont.) Proving logical equivalence: Truth table method A B F1 = A B F3 = (A + B) (A + B) (A + B) 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1