Experiment 3: Logic Simplification


 Kevin Hubert Tate
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1 Module: Logic Design Name:... University no:.. Group no:. Lab Partner Name: Mr. Mohamed ElSaied Experiment : Logic Simplification Objective: How to implement and verify the operation of the logical functions using logic gates. To be familiar with the rules of Boolean algebra to implement the logical functions with minimal representation. To be Familiar with using karnaugh (kmaps) to simplify and optimize the logical functions. Components Required: Mini Digital Training set. IC Type 78 Quadruple input AND gates. IC Type 7 Quadruple input OR gates. IC Type 7 Hex Inverters. Theory: Logical functions: Figure.demonstrates how to implement a simple logical expression using the gates provided. This diagram implements the function f(a,b,c) = A + BC. Since there are three inputs to this function, there are eight possible logical input conditions as shown in the truth table. A B C o/p s (F) A F B C Figure. Reduction techniques: The number of components and connections in a circuit increases exponentially. It is required to examine the logical function to reduce the complexity of the circuit and get simpler circuit design. In this lab, techniques are introduced that can help to implement complex circuits in the least complex manner possible. You can obtain a logically minimal expression by applying the rules of Boolean algebra listed:
2 Complement of a Function: The complement of a function F is F and it is obtained by exchanging s and s in the output column in the truth table. The complement of a function can be derived algebraically by using DeMorgan s laws. (A + B) = A B, and (AB) = A + B for two variables A, B The theorem can be generalized for any number of variables, (A + B + C +. F) = A B C F (A B C.. F) = A + B + C +. F Solved example: Simplify the following Boolean function to a minimum number of literals F = x z + xyz + xz Solution: F = x z + xyz +xz = z (x + x) + xyz = z + xyz = (z + z)(z + xy) = z + xy Truth table: F = z + xy = Σ (,,, 6, 7) minterms x y z F 6 7
3 Kmap simplification: Karnaugh maps or Kmaps provide a graphical technique for simplifying and optimizing logical expressions. The Kmap is best used with logical functions with four or less input variables as follows: Twovariable map Threevariable map Fourvariable map Sum of products and product of sums (SOP & POS): SOP: are boolean functions that formed by SUMMING ANDed terms (minterms). Example: F(A,B,C,D) = A BC+ B D +A C D POS: are boolean functions that formed by PRODUCT ORed terms (maxterms). Example: F(A,B,C,D) = (A+B +C) (B +D ) (A +C+D ) Generally, maxterms are complement of minterms. Solved example: Simplify the Boolean function using Kmap F (x, y, z) = xy + x y z + x yz Solution: F (x, y, z) = xy + x y z + x yz =,,, = Σ (,, 6, 7) F = xy + x z minterms x y z F 6 7
4 Lab part:. Simplify the following Boolean functions to a minimum number of literals. a) F (x,y,z) = Σ (,,,, 6) b) F (A,B,C,D) = (A +C)(A +C )(A+B+C D).. Simplify the following Boolean functions using kmap: c) F (x,y,z) = xy+x y z +x yz d) F (A,B,C,D) = Σ (,,,,, 6, 8, 9,,, ). Procedure: For all the above functions do the following steps: Draw the truth table for that logic function. Simplify the function according to the method of reduction. Draw the logic diagram for the simplified function using basic gates. Implement the simplified function using IC s and breadboard. Give biasing to the IC (i.e. wire the IC to ground (V) and power supply (+ V). With the help of IC s datasheet, connect the input pins of the gate to data switches and the output pin to LED indicator. Verify the truth table for that logic gate and observe the outputs. Lab. Exercise: Students are directed to do the following exercise: Given Boolean function is: F = AB+ ABC+ A C. Write down the function as minterms.. Simplify the function using kmap.. Implement the simplified function using basic logic gates.. Verify the truth table of the simplified function. o/p s (F) After Simplification A B C Actual Observed
5 Part B:. From F get F as Product of sums (POS).. Write down the function as maxterms.. Simplify the function using kmap.. Implement the simplified function using basic logic gates.. Verify the truth table of the simplified function. o/p s (F) After Simplification A B C Actual Observed Comments:
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