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Logic gates and Boolean algebra C.B Pham 2-1 Boolean constants and variables • Boolean constants and variables are allowed to have only two possible values, or • Boolean and not represent actual numbers but instead represent the state of a voltage variable, or what is called its logic level • 0/1 and Low/High are used most of the time C.B Pham 2-2 Truth Tables How a logic circuit output depends on the logic levels present at the inputs C.B Pham 2-3 Logic gates • Three Logic operations: AND, OR, NOT • Logic Gates: Digital circuits constructed from diodes, transistors, and resistors whose output is the result of a basic logic operation (OR, AND, NOT) performed on the inputs C.B Pham 2-4 OR Operation with OR gates Produce a result of whenever any input is Otherwise Truth table C.B Pham Circuit symbol for a two-input OR gate Circuit symbol for a three-input OR gate 2-5 OR Operation with OR gates Determine the OR gate output below: C.B Pham 2-6 AND Operation with AND gates An AND gate output will be only for the case when all inputs are 1; for all other cases the output will be Circuit symbol for a two-input AND gate Truth table C.B Pham Circuit symbol for a three-input AND gate 2-7 AND Operation with AND gates Determine the AND gate output below: C.B Pham 2-8 NOT Operation • The NOT operation is performed on a single variable • Its output logic level is always opposite to the logic level of this input Truth table C.B Pham Circuit symbol for a NOT gate 2-9 Describing logic circuits algebraically Any logic circuit, no matter how complex, can be completely described using basic Boolean operations through AND gates, OR gates, and NOT gates C.B Pham 2-10 Boolean theorems Note: the variable x may represent an expression containing more than one variable C.B Pham AB( AB) AB AB 2-18 Boolean theorems Multivariable theorems C.B Pham 2-19 Boolean theorems - examples Simplify Simplify Simplify C.B Pham 2-20 DeMorgan’s theorems DeMorgan’s theorems are extremely useful in simplifying expressions in which a product or sum of variables is inverted Simplify C.B Pham AC BD 2-21 Universality of NAND gate & NOR gate - Any expression can be implemented using combinations of OR gates, AND gates, and INVERTERs - It is possible to implement any logic expression using only NAND gates (or only NOR gates) and no other type of gate C.B Pham 2-22 Universality of NAND gate & NOR gate C.B Pham 2-23 Universality of NAND gate & NOR gate C.B Pham 2-24 Universality of NAND gate & NOR gate Example: implement the logic circuit for x = AB + CD with a minimum number of ICs The TTL integrated circuits are available Each IC is a quad C.B Pham 2-25 Universality of NAND gate & NOR gate C.B Pham 2-26 Alternate logic gate representations It is common to find circuit diagrams that utilize alternate logic symbols in addition to the standard symbols C.B Pham 2-27 Alternate logic gate representations C.B Pham 2-28 Alternate logic gate representations • Logic symbol interpretation: active-HIGH / active-LOW C.B Pham 2-29 Alternate logic gate representations Proper use of the alternate gate symbols can make the circuit operation much clearer ? Truth table C.B Pham 2-30 Alternate logic gate representations C.B Pham 2-31 Alternate logic gate representations Whenever possible, choose gate symbols so that bubble outputs are connected to bubble inputs, and nonbubble outputs to nonbubble inputs C.B Pham 2-32
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