ME 140L - Lecture 6 - Combinatorial Networks - Karnaugh Mapping

ME 140L - Lecture 6 - Combinatorial Networks - Karnaugh Mapping

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1 LOGIC GATES & COMBINATORIAL CIRCUITS Consider a vending machine: 1. You wish to make a selection and note the cost 2. You don’t have the exact change 3. You put in the coins (dollar) and press the appropriate snack selection 4. How does the machine determine whether or not to dispense the snack?
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2 MECHATRONICS LAB ME 140L COMBINATORIAL CIRCUITS AND KARNAUGH MAPPING I. BOOLEAN ALGEBRA AND LOGIC DEVICES A. BOOLEAN ALGEBRA Mathematical structure consisting of a set containing 0 and 1: complement, addition and multiplication. Complement AND OR Exclusive OR A A A B AB A B A+B A B A B 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 Multiplication (Product) Addition (Sum) Note: Product is 1 iff all terms are1 and Sum is 0 iff all terms are 0. Product is 0 iff one term is 0 Boolean Equivalences (needed to reduce the complexity of expressions) B A B B A A B) A(A BC A C) B)(A (A A ) B B)(A (A AB )B B (A A AB A A B A AB Laws s Morgan' De B A AB B A B) (A CA BA C)A (B AC AB C) A(B C B) (A C) (B A (assoc) A(BC) (AB)C (comm) A B B A ) (comm BA AB 1 A A 0 A A 1 1 A A 1 A A 0 A 0 0 A A A A A AA A A + = + = + + = + + = + + = + = + = + + = = + + = + + = + + + = + + = + = + = = + = = + = = + = = + = =
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3 LOGIC DEVICES (3-input shown) AND OR XOR NOT NAND NOR XNOR
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4 II. COMBINATORIAL NETWORKS Combinations of binary inputs to generate a desired binary output Example: Two-to-Four Decoder ) 4 2 ( 2 = Two inputs: A B (each 0 or 1) Four possible outputs: x 1 , x 2 , x 3 , x 4 2 to 4 Decoder x 1 x 2 x 3 x 4 A B For each combination of A and B, only one x i goes high: called Truth Table A B Output 0 0 x 1 0 1 x 2 1 0 x 3 1 1 x 4 Use ANDs and Inverters only to implement a combinatorial circuit that will mimic the 2-to-4 decoder x3 x1 x2 x4 B A Use NANDs and Inverters only B A x1 x2 x4 x3
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ME 140L - Lecture 6 - Combinatorial Networks - Karnaugh Mapping

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