Derived logic gates

Transcript

1. DERIVED LOGIC GATES Symbols • Boolean Expressions • Truth Tables • Universal Gates Shrushti Jatinder Kadam Assistant Professor ECE Department
2. DERIVED LOGIC GATES Symbols • Boolean Expressions • Truth Tables • Universal Gates NAND NOR XOR / XNOR NOT + AND NOT + OR Different / Same
3. Learning Objectives Objective 1 Objective 2 Define derived logic gates. Write expressions and identify symbols for NAND, NOR, XOR and XNOR. Objective 3 Objective 4 Construct and interpret their truth tables. Compare the four gates using output conditions. Objective 5 Objective 6 Explain why NAND and NOR are universal gates. Implement NOT, AND and OR using NAND/NOR. 2
4. Prerequisite: Basic Logic Gates A quick reminder before studying derived gates AND OR NOT Output = 1 only when all inputs are 1. Output = 1 when at least one input is 1. Output is the complement of the input. Y=A·B Y=A+B Y = A′ 3
5. What Are Derived Logic Gates? Logic functions formed by combining or complementing basic operations Core idea NAND Derived gates extend the basic AND, OR and NOT operations. Y = (A·B)′ Common derived gates: • NAND • NOR • XOR • XNOR NOR Y = (A+B)′ XOR Y=A⊕B XNOR Y = (A ⊕ B)′ 4
6. NAND Gate NAND = NOT-AND A B Y A B Y 0 0 1 Output is 0 only when both inputs are 1. 0 1 1 Memory tip: NAND is the complement of AND. 1 0 1 1 1 0 Key idea Y = (A · B)′ 5
7. NOR Gate NOR = NOT-OR A B Y A B Y 0 0 1 Output is 1 only when both inputs are 0. 0 1 0 Memory tip: NOR is the complement of OR. 1 0 0 1 1 0 Key idea Y = (A + B)′ 6
8. XOR Gate Exclusive-OR: output is 1 when inputs are different A A B Y 0 0 0 0 1 1 1 0 1 1 1 0 Y B Key idea For two inputs, XOR produces 1 for 01 and 10. Think: inputs are different. Y=A⊕B 7
9. XNOR Gate Exclusive-NOR: output is 1 when inputs are the same A B Y 0 0 1 XNOR is the complement of XOR. 0 1 0 For two inputs, output is 1 for 00 and 11. 1 0 0 1 1 1 Key idea Y = (A ⊕ B)′ 8
10. Quick Comparison The fastest way to remember the four gates NAND NOR XOR XNOR 0 only for 11 1 only for 00 1 for different inputs 1 for same inputs Key relationship XOR and XNOR are complements: XNOR = (XOR)′. 9
11. Universal Gates NAND and NOR can each realize a complete set of basic logic functions NAND as a universal gate NOR as a universal gate Using only NAND gates, we can implement: Using only NOR gates, we can implement: • NOT • AND • OR • NOT • OR • AND Therefore, any Boolean function can be constructed using NAND gates. Therefore, any Boolean function can be constructed using NOR gates. Universal gate → complete logic implementation 10
12. NAND as NOT Step-by-step Boolean implementation Y = (A·A)′ = A′ Connection Connect both inputs of one NAND gate to A. Operation A·A = A. Result Y = A′, so one NAND implements NOT. 11
13. NAND as AND Step-by-step Boolean implementation Y = ((A·B)′)′ = A·B Stage 1 X = (A·B)′ using NAND. Stage 2 Feed X to both inputs of a second NAND. Result Y = A·B. 12
14. NAND as OR Step-by-step Boolean implementation Y = (A′·B′)′ = A+B t Stage 1 Use two NAND gates as NOT gates to get A′ and B′. Stage 2 Feed A′ and B′ to a third NAND. Result By De Morgan’s law, Y = A+B. 13
15. NOR as NOT Step-by-step Boolean implementation Y = (A+A)′ = A′ Connection Connect both inputs of one NOR gate to A. Operation A+A = A. Result Y = A′. 14
16. NOR as OR Step-by-step Boolean implementation Y = ((A+B)′)′ = A+B Stage 1 X = (A+B)′ using NOR. Stage 2 Feed X to both inputs of a second NOR. Result Y = A+B. 15
17. NOR as AND Step-by-step Boolean implementation Y = (A′+B′)′ = A·B Stage 1 Use two NOR gates as NOT gates to get A′ and B′. Stage 2 Feed A′ and B′ to a third NOR. Result By De Morgan’s law, Y = A·B. 16
18. Applications Where these gates appear in digital systems XOR XNOR NAND / NOR • Binary addition • Parity generation/checking • Data comparison • Equality detection • Digital comparators • Error-checking logic • Control logic • Memory/sequential circuits • General Boolean implementation Design Insight Universal gates are useful because a complete logic system can be built from a single gate family. 17
19. THANK YOU Keep practicing Boolean expressions and truth tables. Next: Boolean Algebra • Universal Gates • Combinational Circuits