EXPERIMENT NO.1:INTRODUCTION TO BASIC GATES AND LOGIC SIMPLIFICATION TECHNIQUES

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1 DEPARTMENT OF ELECTRICAL AND ELECTROINC ENGINEERING BANGLADESH UNIVERSITY OF ENGINEERING & TECHNOLOGY EEE 304 : Digital Electronics Laboratory EXPERIMENT NO.1:INTRODUCTION TO BASIC GATES AND LOGIC SIMPLIFICATION TECHNIQUES Prelab work: Submit your Prelab before starting the experiment 1. Form the truth table for three basic gates known as AND, OR and NOT Gate (Inverter). Assume the number of inputs to be 3. Required Instruments andics 1) 2 input AND gate (74LS08) 2) 2 input OR gate (74LS32) 3) Not gate (74LS04) Part A: introduction to most familiar gates: While the three basic functions AND, OR, and NOT are sufficient to accomplish all possible logical functions and operations, some combinations are used so commonly that they have been given names and logic symbols of their own. The NAND Gate The NAND gate implements the NAND function, which is exactly inverted from the AND function you already examined. With the gate shown in figure, both inputs must have logic 1 signals applied to them in order for the output to be a logic 0. With either input at logic 0, the output will be held to logic 1. X Y Z=(X.Y) / The circle at the output of the NAND gate denotes the logical inversion.also in the figure, note that the overbar is a solid bar over both input values at once. This shows that it is the AND function itself that is inverted, rather than each separate input. The NOR Gate The NOR gate is an OR gate with the output inverted. Where the OR gate allows the output to be true (logic 1) if any one or more of its inputs are true, the NOR gate inverts this and forces the output to logic 0 when any input is true. X Y (X+Y) /

2 In symbols, the NOR function is designated with a plus sign (+), with an overbar over the entire expression to indicate the inversion. In logical diagrams, this is indicated by an OR gate with a circle to designate the inversion. The Exclusive-OR, or XOR Gate The Exclusive-OR, or XOR function is an interesting and useful variation on the basic OR function. Verbally, it can be stated as, "Either A or B, but not both." The XOR gate produces a logic 1 output only if its two inputs are different. If the inputs are the same, the output is a logic A B AB The XOR symbol is a variation on the standard OR symbol. It consists of a plus (+) sign with a circle around it. The logic symbol, as shown here, is a variation on the standard OR symbol. EXPERIMENT 1: Implement the following logic function with basic gates F(A,B,C) = AC+BC Write down the truth table and compare it with your experimental results. Part B : Boolean Minimization_- an algebraic approach_ One of the primary requirements when dealing with digital circuits is to find ways to make them as simple as possible. This constantly requires that complex logical expressions be reduced to simpler expressions that nevertheless produce the same results under all possible conditions. The simpler expression can then be implemented with a smaller, simpler circuit, which in turn saves the price of the unnecessary gates, reduces the number of gates needed, and reduces the power and the amount of space required by those gates. One tool to reduce logical expressions is the mathematics of logical expressions, introduced by George Boole in 1854 and known today as Boolean Algebra. The rules of Boolean Algebra are simple and straight-forward, and can be applied to any logical expression. The rules of Boolean Algebra are: 2

3 AND Operations ( ) 0 0 = 0 A 0 = = 0 A 1 = A 0 1 = 0 A A = A 1 1 = 1 A A' = 0 OR Operations (+) 0+0 = 0 A+0 = A 1+0 = 1 A+1 = = 1 A+A = A 1+1 = 1 A+A' = 1 NOT Operations (') 0' = 1 1' = 0 A'' = A Associative Law (A B) C = A (B C) = A B C (A+B)+C = A+(B+C) = A+B+C Commutative Law A B = B A A+B = B+A Distributive Law A (B+C) = (A B) + (A C) A+(B C) = (A+B) (A+C) Precedence AB = A B A B+C = (A B) + C A+B C = A + (B C) DeMorgan's Theorem (A B)' = A' + B' (NAND) (A+B)' = A' B' (NOR) Example:A given Boolean expression ( AB + AC + ABC ) can be solved by following methods: STEP1:Apply DeMorgan s theorem to the first term ( AB) ( AC) + A B C STEP2:Apply DeMorgan s theorem to each term in parentheses ( A + B)( A + C) + A B C STEP3:Apply the distributive law to the two terms in parenthesis A A + A C + A B + B C + A B C STEP4:Apply the AND operation( A A ( A B(1 + C) = A B )to the third and last term. A + A C + A B + B C = A ) to the first term and OR operation STEP5: Apply OR operation ( A (1 + C) = A ) to the first and second terms A + A B + B C STEP5: Apply OR operation to the first and second terms A + B C Problem: For the following logic function, find out the truth table, write down the logic expression and simplify the logic expression as far as possible using Boolean algebra. F(A,B,C,D)= (5,7,10,11,13,14,15) 3

4 Draw the circuit diagrams both before and after simplification. Can you see the effect of simplification? Are you sure that further simplification is not possible???? Is your simplified logic expression matches exactly with that of all your classmates????? Part C: Karnaugh Maps(or in short k map): A Geometric Approach K-Map: a truth table arranged so that terms which differ by exactly one variable are adjacent to one another so we can see potential reductions easily. RULES: 1)A group or sub-cube must contain either 1,2,4,8 or 16 cells(i.e 2 N cells ). 2) Each cell in a group must be adjacent to one or more cells in that same group, but all cells in the group do not have to be adjacent to each other. 3) Always include the largest possible number of 1s in a group in accordance with rule 1. 4) Each 1 on the map must be included in at least one group. The 1s already in a group can be included in another group as long as the overlapping groups include noncommon 1s. 4

5 RULES: 1.Group the cells that have 1s.Each group of cells containing 1s creates one product term composed of all variables that occur in only form(either uncomplemented or complemented within the group are eliminated) 2.Determine minimum product terms for each group 3.When all the minimum product terms are derived from the Karnaugh map, they are summed to form the minimum SOP expression EXPERIMENT 2: (a) Simplify the problem as shown in Part B using K map and draw and construct the circuit diagram. Which method provides better simplification? Implement your simplified circuit in the breadboard and verify the functionality of the circuit. (b) Simplify the following problem using K map. Draw and construct the circuit diagram F(A,B,C,D)=Σ(5,6,7,9,13) Is there only one solution or more? Report: 1) For each of the experiment above, formulate the problem, show the implemented circuit diagram and the measured truth table of your circuit. 5

6 2) Answer all the questions in the sheet including the additional questions asked below. 3) Discussion Additional questions: 1. It is possible to implement all the basic logic gates using only NAND gate. That is why it is called universal gate. With necessary circuit diagram,show how AND, OR,NOT and XOR gates can be implemented using NAND gates only. 2. Simplify(circuit diagram not required) the following problem using K map.(note that it is a 5 variable problem) F(A,B,C,D,E)=Σ(1,4,5,13,20,21,22,28) +d(6,9,11,12,14,29,30) Here, D means 'DONT CARE' IC PIN Diagram IC7408/ INPUT AND/OR GATE INPUT OUTPUT GND 14 VCC IC 7404 INVERTER INPUT OUTPUT GND Reviewed By: Md. Imran Momtaz 6