Lecture #9 - Karnaugh Maps

# Lecture #9 - Karnaugh Maps - ECE 301 – Digital...

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Unformatted text preview: ECE 301 – Digital Electronics Karnaugh Maps (Lecture #9) Learning Objectives • Karnaugh Maps – 2-variable – 3-variable – 4-variable • How to use K-Maps – Simplifying a minterm expansion (minimum SOP) – Simplifying a maxterm expansion (minimum POS) Spring 2012 2 ECE 301 - Digital Reading Roth & Kinney Sections 5.1 – 5.3 Spring 2012 ECE 301 - Digital 3 Simplification • Boolean Algebra is useful for general proofs. • And can be used to simplify logic functions. • However, two problems arise 1. Algebraic procedures are difficult to apply in a systematic way. 2. Difficult to determine when a minimum solution has been achieved. • Karnaugh maps are generally faster and Spring 2012 ECE 301 - Digital 4 Simplification: Example Find the minimum SOP expression for the following logic function: F(A,B,C) = Σ m(0,1,2,5,6,7) Spring 2012 ECE 301 - Digital 5 Simplification: Example F(A,B,C) = m0 + m1 + m2 + m5 + m6 + m7 F(A,B,C) = A’B’C’ + A’B’C + A’BC’ + AB’C + ABC’ + ABC F = (A’B’C’ + A’B’C) + (A’B’C + AB’C) + (A’BC’ + ABC’) + (ABC’ + ABC) F = A’B’ + A’C + BC’ + AB Spring 2012 ECE 301 - Digital 6 Is this the minimum SOP expression? Simplification: Example F(A,B,C) = m0 + m1 + m2 + m5 + m6 + m7 F(A,B,C) = A’B’C’ + A’B’C + A’BC’ + AB’C + ABC’ + ABC F = (A’B’C’ + A’B’C) + (A’BC’ + ABC’) + (AB’C + ABC) F = A’B’ + BC’ + AC Spring 2012 ECE 301 - Digital 7 Unfortunately, it is not! Instead, Karnaugh Maps Like a truth table, the Karnaugh map of a function specifies the logical value of the function for all combinations of the input variables. Spring 2012 ECE 301 - Digital 8 Two-variable K-map Spring 2012 ECE 301 - Digital 9 Row # A B minterm m0 1 1 m1 2 1 m2 3 1 1 m3 m0 m2 m1 m3 1 1 A B m0 = A’B’ m1 = A’B m2 = AB’ m3 = AB A = 0, B = 1 A = 1, B = 0 Two-variable K-map Spring 2012 ECE 301 - Digital 10 Row # A B maxterm M0 1 1 M1 2 1 M2 3 1 1 M3 M0 M2 M1 M3 1 1 A B M0 = A + B M1 = A + B’ M2 = A’ + B M3 = A’ + B’ Two-variable K-map: Example Spring 2012 ECE 301 - Digital 11 Row # A B F 1 1 1 2 1 1 3 1 1 1 1 1 1 A B Minterm Expansion: F(A,B) = A’B’ + AB’ = Σ m(0, 2) Maxterm Expansion: F(A,B) = (A’ + B).(A’ + B’) = Π M(1, 3) 2 1 3 Can this logic function be simplified?...
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Lecture #9 - Karnaugh Maps - ECE 301 – Digital...

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