218L5F08a - ESE218 Lecture 5. Switching algebra. Two-level...

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9/16/08 ESE218 Fall 2008 Lecture 5 1 ESE218 Lecture 5. Switching algebra. Two-level implementations. Standard forms Outline ± Switching algebra ² Positive and negative logic ² Duality principle ² Axioms ² Distributive law ² Theorems ² DeMorgan’s law ± Functionally complete sets ± NAND ± NOR ± XOR and XNOR operations ± Functions of two variables and extension to multiple inputs ± Circuit levels and propagation delay ± Standard forms ² sum of products ² products of sums ² form conversion ± Summary
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9/16/08 ESE218 Fall 2008 Lecture 5 2 Switching algebra Claude Elwood Shannon 1916-2001 B = { 0, 1 } (X) OR (Y) = X + Y (X) AND (Y) = X Y NOT (X) = X’
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9/16/08 ESE218 Fall 2008 Lecture 5 3 Positive and negative logic +5 +5 +5 +5 GND +5 +5 +5 GND GND GND GND F Y X GND = 0 +5 = 1 GND = 1 +5 = 0 1 1 1 1 0 1 1 1 0 0 0 0 F Y X 0 0 0 0 1 0 0 0 1 1 1 1 F Y X F = X + Y F = X Y Assuming positive logic Assuming negative logic V1 5Vdc 1k R1 0 4 3 1 2 4 3 1 2 F Y X
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9/16/08 ESE218 Fall 2008 Lecture 5 4 Duality principle Every algebraic expression remains valid if (1) operators “+” and “ (2) identity elements “0” and “1” are interchanged: Change 0 1 1 0 + →⋅ ⋅→ + Valid Expression (X + 1) 0 = 0 Valid Expression X 0 + 1 = 1 Keep the original order of operations!
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9/16/08 ESE218 Fall 2008 Lecture 5 5 Functionally complete sets 5 6 4 U4B 7402 4 5 6 U3B 7400 1 2 3 U1A 7400 F = (X+Y)’ X Y F = (XY)’ X Y X 4 5 6 U1B 7400 9 10 8 U1C 7400 12
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218L5F08a - ESE218 Lecture 5. Switching algebra. Two-level...

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