LAB 2: BOOLEAN THEOREMS

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1 LAB 2: BOOLEAN THEOREMS OBJECTIVES 1. To implement DeMorgan's theorems in circuit simplification. 2. To design a combinational logic circuit with simplest logic gates representation using Karnaugh Mapping Technique. EQUIPMENTS/COMPONENTS Quarters II INTRODUCTION Theorems of Boolean Algebra are a set of rules used with digital variables and logical operations to develop, manipulate and simplify logical expressions. Boolean Algebra provide systematic means for discovering alternative expressions. This lab session will let you get familiarize with De Morgan s and Karnaugh Technique. De Morgan Theorems De Morgan's theorem allows large bars in a Boolean expression to be broken up into smaller bars over individual variables. De Morgan's theorem says that a large bar over several variables can be broken between the variables if the sign between the variables is changed. From the above equation, it is clear that a NAND gate is equal to an OR gate with inverted inputs and a NOR gate is equal to an AND gate with inverted inputs. In order to reduce expressions with large bars, the bars must first be broken up. This means that in some cases, the first step in reducing an expression is to use De Morgan's theorem. De Morgan's theorem is useful in the implementation of the basic gate operations using alternative gates, particularly operations involving NAND and NOR gates. For this session, you will implement De Morgan Theorem into a given Boolean Equation. Karnaugh Maps Karnaugh Map provides a systematic method for simplifying Boolean expressions. Using K-Map is very similar to truth table. K-Map can be used for expressions with two, three, four or five variables.

2 Number of cells in a K-Map is equivalent to the number of possible input variable combinations (2 n ). As an example: For 3-variables; number of cells are 2 3 = 8 The cells in K-Map are arranged so that only one single variable changes between adjacent cells. In the experiment, you will be given a design problem. You should be able to interpret the design problem into a truth table and apply the Karnaugh Map technique to find the simplest logic expression. Finally, you have to construct and test the circuit. PROCEDURE PART A : Simplification using De Morgan's theorems 1. Draw a logic diagram for the equation : Y = AB B+C 2. Construct the circuit using the diagram you drew in Step 1. Connect toggle switches to inputs A, B and a LED to the circuit output, Y. Set the toggle switches to each input combination listed in Table 2.1, and record the output value observed in the table.

3 INPUTS OUTPUT Table 2.1 : Step 1 Circuit Operation 3. Apply De Morgan s laws to remove the top inversion bar by changing the sign. Get the simplified expression and draw the logic circuit diagram in the space given below:

4 4. Construct a logic circuit for the simplified expression obtained in step 3 and again complete the truth table in Table 2.2. INPUTS OUTPUT Table 2.2 : Step 3 Circuit Operation 5. Write your observation based on the results.

5 PART B: Design Problem using Karnaugh Mapping Technique A jet aircraft employs a system for monitoring the rpm, pressure, and temperature values of its engines using sensors that operate as follows: RPM sensor output = 0, only when speed < 4800 rpm P sensor output = 0, when pressure <220 psi T sensor output = 0, only when temperature < 200 F Figure 2.1 shows the logic circuit that controls a cockpit warning light for certain combinations of engine conditions. Assume that a HIGH at output W activates the warning light: (a) Create your truth Table with appropriate input-output combinations (b) Determine what engine conditions will give warning to the pilot. (c) Using K-Map technique to obtain your simplified Boolean equation. (d) Change this circuit to one using all NAND gates and constructs the circuit. Figure 2.1 REPORT REQUIREMENT The lab report should include the following: 1. Summary of each experiment done. 2. Results of each experiment. 3. Discussions on each operation of the gates. 4. For Activity Part B, include your K-Map table, the simplified Boolean equation, the circuit diagram, the truth table and the engine conditions. 5. Remarks on any possible failures and measures done to overcome this problem. 6. Conclusion for the experiments.

6 PGT 104 DIGITAL ELECTRONICS EXPERIMENT DATA SHEET NAME : METRIC NO.: COURSE : LAB EXPERIMENT : PART A : Simplification Using De Morgan Theorem equivalent operation Y = inputs outputs inputs outputs Table 2.1 : Step 1 Opertaion Table 2.2 : Step 3 Operation INSTRUCTOR S SIGNATURE:

7 PART B : Karnaugh Map Design Method (a) INPUTS OUTPUT Truth Table (b) The engine conditions that will give warning to the pilot: (c) K-Mapping : Simplified Boolean equation:

8 (d) Logic Circuit Diagram using all NAND gates INSTRUCTOR S SIGNATURE: