Lecture 5 - Th The University of Texas at Dallas Erik...

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Erik Jonsson School of Engineering and h U i it f T t D ll Computer Science The University of Texas at Dallas Simplifying Logic Circuits with Karnaugh Maps The circuit at the top right is the logic equivalent of the Boolean expression : f = a b c + a b c + a b c ow as we have seen this expression Now, as we have seen, this expression can be simplified (reduced to fewer terms) from its original form, using the Boolean identities as shown at right. he circuit may be simplified as The circuit may be simplified as follows: f = abc + abc + abc = f = (abc + abc) + abc + abc (since x = x + x), f = (abc + abc) + (abc + abc), or 1 1 f = ac (b + b) + ab (c + c), so that f = ab + ac. The simplified circuits is © N. B. Dodge 09/09 Lecture #5: Logic Simplification Using Karnaugh Maps 1 shown bottom right.
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Erik Jonsson School of Engineering and h U i it f T t D ll Computer Science The University of Texas at Dallas Simplifying Logic Circuits (2) Since you have now had some experience with simplification of Boolean expressions, this example is (hopefully) familiar and understandable. However, for more complex Boolean expressions, the identity/substitution approach can be VERY cumbersome (at Original logic circuit ( least, for humans). Instead of this approach, we can use a graphical technique called © N. B. Dodge 09/09 Lecture #5: Logic Simplification Using Karnaugh Maps 2 the Karnaugh map . Simplified equivalent logic circuit
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Erik Jonsson School of Engineering and h U i it f T t D ll Computer Science The University of Texas at Dallas Karnaugh Maps Another approach to simplification is called the Karnaugh map , or K-map. A K-map is a truth table graph , which x y y 00 01 aids in visually simplifying logic. It is useful for up to 5 or 6 variables, and is a good tool to help understand the process of logic simplification. x 2 3 10 11 0 1 This minterm is expressed as f=xy. The algebraic approach we have used previously is also used to analyze complex circuits in industry (computer analysis) . t the right is a 2 ariable K ap Two-Variable K-map , labeled for SOP terms. Note the four squares At the right is a 2-variable K-map. This very simple K-map demonstrates that an n-variable K-map contains all the combination of the n variables in the K- represent all the com- binations of the two K-map variables, or interms in x & y © N. B. Dodge 09/09 Lecture #5: Logic Simplification Using Karnaugh Maps 3 map space . minterms , in x & y (example above).
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Erik Jonsson School of Engineering and h U i it f T t D ll Computer Science The University of Texas at Dallas Three-Variable Karnaugh Map A useful K-map is one of three variables. Each square represents a 3-variable minterm or maxterm. yz yz yz yz –– All of the 8 possible 3-variable terms are represented on the K-map.
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This note was uploaded on 10/16/2009 for the course EE 2310 taught by Professor Dodge during the Spring '09 term at University of Texas at Dallas, Richardson.

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Lecture 5 - Th The University of Texas at Dallas Erik...

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