Having read this workbook you should be able to: design a logic circuit from its Boolean equation or truth table.
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1 Objectives Having read this workbook you should be able to: analyse a given logic circuit by deriving its oolean equation and completing its truth table. design a logic circuit from its oolean equation or truth table. draw logic diagrams showing how NND gates can be used to make up NOT, ND, OR, NND and EXOR gates. reduce a given logic circuit down to NND gates only. 1
2 OOLEN EUTION FOR LOGIC CIRCUIT In Chapter 1 you discovered how symbols (operators) are used to represent the function of NOT, ND, OR, NND, NOR and EXOR logic gates. a dot. is used as an operator to represent ND, a plus sign + is used to represent OR, a bar, on top of the variables, is used to represent NOT, a + is used to represent exclusive OR (EXOR). The following table should remind you of how the operators are used to represent the functions of logic gates. GTE SYMOL OOLEN EUTION NOT = ND =. OR = + NND =. NOR = + EXOR = + EXNOR = + 2 Logic gates are often combined together to provide a logic function not available from a single type of gate.
3 NLYSING LOGIC CIRCUIT EXMPLE 1 We shall now consider examples of how oolean algebra can be used to analyse a given system. Full guidance is provided with the first example. Derive an equation, and form a truth table, representing the logic function of the following circuit. = +.. We start by following the inputs to where they enter gates and labelling these points. Provided all input points are labelled, we can use oolean notation to label the output. The output of the NOT gate is labelled, and the output of the ND gate labelled.. These can now be followed through to the OR gate. The OR gate has at one of its inputs and. at the other. The output of the OR gate is given by: = +. The oolean equations tells us that is at logic level 1 when: = 0 (the in the equation) OR when: = 1 ND = 1 (the. in the equation) We can now complete a truth table for the logic arrangement Covers situations where = Covers = 1 ND = 1 Later on you will see how the rules of oolean algebra can be used to simplify a circuit i.e. how to provide the same function using fewer gates. 3
4 EXMPLE 2 Derive the oolean equation and form a truth table which represents the function of the following logic system.. C (a) Try using oolean algebra to label your way from the inputs to the output. You should be able to show that the output can be represented in terms of the inputs, and C by the equation: = C The equation tells us that the output will be at logic level 1 when: = 0 ND = 1 (the. in the equation) OR when: = 1 ND = 0 (the. in the equation) OR when: = 1 ND = 1 (the. in the equation) OR when: C = 1 (b) Now place 1's in the output column of the following truth table, where required, for each of the four conditions. The. term has been done for you. 1 has been inserted in the output column where ( = 0, = 1, C = 0) and where ( = 0, = 1, C = 1) 4
5 s you work your way through the conditions producing an output at logic level 1, you will often find that the previous condition has already filled the space. This suggests that there is an overlap between the conditions and that the system can be simplified. Simplifying techniques will be covered in Chapters 3 and 4. C Cover (c) Fill any empty output boxes with a 0. If you have successfully completed the task there should only be one box with a 0 in it. Study the truth table carefully and try to decide what single 3-input logic gate could be used to produce the same function. complete solution is provided on page 12 of this chapter. 5
6 DESIGNING LOGIC CIRCUITS The following sequence illustrates the steps involved in designing a logic circuit. form a truth table representing the required logic function; derive the oolean equation from the truth table; if possible, simplify the equation; draw a logic diagram showing how the circuit could be made up from ND, OR and NOT gates; reduce the system down to NND gates only; Let us now consider these steps in some detail. FROM TRUTH TLE TO OOLEN EUTION Suppose that the function of a logic circuit could be represented by the following truth table. C C C C The oolean equation covers situations when the output is at logic level 1. In this case, output is a 1 when: = 1, ND = 0, ND C = 0 (represented by..c) OR when: = 1, ND = 1, ND C = 0 (represented by..c) OR when: = 1, ND = 1, ND C = 1 (represented by..c) 6
7 The full oolean equation for the circuit becomes: =..C +..C +..C It will be shown in the next Chapter that the equation simplifies down to: =.( + C) FROM OOLEN EUTION TO LOGIC CIRCUIT DIGRM The oolean equation for the circuit is: =.( + C) We start by drawing a logic circuit diagram for the term inside the bracket. This part of the circuit must produce an output at logic level 1 when: is at logic level 1, OR when: C is NOT at logic level 1. suitable arrangement is shown below. C C ( + C) To complete the equation, ( + C) is NDed with input. The complete circuit diagram becomes: C Note that the complete contents of the brackets are NDed with input. 7
8 SOLUTION FOR EXMPLE 2 (Page 6) C.. C C rrangement forms a 3-input OR gate. 8
9 Student ssessment The following equations are used with questions 1 and 2. =. + C =. + C C =. + C D =. + C 1. C Which oolean equation represents the function of the logic circuit shown above? 2. C Which oolean equation represents the function of the logic circuit shown above? 9
10 The following truth tables are used with questions 3 and 4. C C C C D C Which truth table represents the function of the logic circuit in question 1? 4. Which truth table represents the function of the logic circuit in question 2? The following equations are used with questions =. =. +. C =. D = + E = + F =
11 5. Which equation applies to this truth table? Which equation applies to this truth table? Which equation applies to this truth table? Which equation applies to this truth table? Which equation applies to this truth table? Which equation applies to this truth table?
12 Exercise Objectives Having complete this Exercise you should be able to: set up a logic circuit and complete its truth table. construct 2-input ND, OR, NOR and EXOR gates using only NND gates. reduce a logic arrangement made up of different gates down to NND gates only. 12
13 CTIVITY 1 1a. Set up the following logic gate arrangement: CTIVITY 2 1b. Use the arrangement to enable you to complete the truth table. 2a. Set up the following logic gate arrangements: b. Use the arrangement to enable you to complete the truth table
14 CTIVITY 3 Let us now consider how we can make other types of gates from NND gates only. 3a. Set up the following arrangements, in turn and complete their truth table. 3b. INPUT 0 1 OUTPUT 3c d
15 3e f
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