Logic Circuit Design

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Claude Bates
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1 Logic Circuit Design

2 we have studied Truth Tables Logic gates Logic algebra K-maps 1

3 All these are tools Tools Truth Tables Logic gates Logic algebra K-maps 2

4 All these are tools Tools Truth Tables Logic gates Logic algebra K-maps To design digital logic circuits 3

5 Today we will learn Truth Tables Logic gates Logic algebra K-maps To set-up and implement logic circuit problems (Logic Circuit Design) 4

6 4-Questions for a design problem 1. What do we need to know for a design problem? Specifications (inputs, outputs, function) 2. Can we set-up a truth table? Truth tables determine the function of the problem 3. Can we simplify? - Use K- maps or logic algebra 4. Can we implement the result with a circuit? - Use gates 5

7 Steps to Design a Logic Circuit Given a word problem 6

8 Steps to Design a Logic Circuit Given a word problem Create Truth Table 7

9 Steps to Design a Logic Circuit Given a word problem Create Truth Table Derive Output Expression(s) 8

10 Steps to Design a Logic Circuit Given a word problem Logic Circuit Create Truth Table Derive Output Expression(s) 9

11 Steps to Design a Logic Circuit Given a word problem Logic Circuit Create Truth Table Derive Output Expression(s) Simplify 10

12 Steps to Design a Logic Circuit Given a word problem Logic Circuit Create Truth Table Derive Output Expression(s) Simplify Implement (Gates) 11

13 Algorithm: Logic circuit design 1. From the specifications of the problem: Determine the required number of inputs and outputs. Assign a letter symbol to each 2. Derive the truth table 3. Obtain the Boolean expressions for each output as a function of the input variables 4. Simplify all the output equations 5. Draw the logic diagram 6. Verify the correctness of the design 12

14 Algorithm: Logic circuit design S y n t h e s I s 1. From the specifications of the problem: Determine the required number of inputs and outputs. Assign a letter symbol to each 2. Derive the truth table 3. Obtain the Boolean expressions for each output as a function of the input variables 4. Simplify all the output equations 5. Draw the logic diagram Implementation 6. Verify the correctness of the design 13

15 Example: Design problem Design a digital logic circuit with three inputs and one output. The output of the logic circuit, must be logic one(1)2 when the binary value of the inputs is less than three(011)2 and zero(0)2 otherwise. 14

16 1. Inputs/Outputs Inputs = 3 Output = 1 15

17 Inputs/Outputs/Truth table Inputs = 3 Output = 1 Design a digital logic circuit with three inputs and one output. The output of the logic circuit, must be logic one(1)2 when the binary value of the inputs is less than three(011)2 and zero(0)2 otherwise. A B C X

18 2. Function - Truth table A B C X Design a digital logic circuit with three inputs and one output. The output of the logic circuit, must be logic one(1)2 when the binary value of the inputs is less than three(011)2 and zero(0)2 otherwise

19 Terms of the output expression A B C X ABC ABC ABC 18

20 3. Output Boolean expression _ X = ABC + ABC + ABC A B C X ABC ABC ABC 19

21 4. Simplification K-Map AB C ABC ABC ABC 20

22 4. Simplification K-Map AB C ABC ABC ABC 10 X = A B + A C 21

23 5. Implementation: Simplified logic circuit A = 1 B = 0 C = 1 X = A B + A C 22

24 6. Verification A = 1 B = 0 C =

25 Logic diagram-vhdl editor Not in the Exam 26

26 Design: 1-bit Binary Comparator Compare two binary digits New Example 29

27 Design: 1-bit Comparator A 1-bit binary comparator compares the values of two bits and produces the proper result. A B A, B = 0 or 1 30

28 Design: 1-bit Comparator An 1-bit binary comparator compares the values of two bits and produces the proper result. What is the proper result? 31

29 1-bit Comparator There are three cases: 1. The two bits are equal 2. One bit has a greater value that the other 3. One bit has a smaller value than the other 32

30 Inputs/Outputs/Function Inputs:? Outputs:? Function: Comparator 33

31 Inputs/Outputs/Function Inputs: 2 Outputs: 3 Function: Comparator 34

32 Label the Input/Output A B? E (A = B) L (A < B) G (A > B) 35

33 Truth table? A B A = B A < B A > B E L G 36

34 Truth table A B A = B A < B A > B E L G 37

35 Truth table E L G A B A = B A < B A > B

36 We have 3 output equations E L G A B A = B A < B A > B E = A B + AB L = A B G = AB 39

37 1-bit Comparator (Logic Circuit) E = A B + AB L = A B G = AB 40

38 VHDL: Logic circuit E = A B + AB = A XNOR B G = AB L = A B 41

39 VHDL Not in the Exam 42

40 Another design example Design a BCD-to-7 Segment Display converter New Example 43

41 Input, outputs(s), function? BCD 7SD (7-Segment-Display) 44

42 BCD BCD = Binary Coded Decimal A B C D

43 7SD 46

44 7SD 47

45 BCD to 7SD converter A B C D BCD to 7-segment decoder 48

46 49

47 Simplify 50

48 K-maps 51

49 Simplified equations 52

50 Implement VHDL 53

51 Implement using a chip (7446) 54

52 Design a binary converter Design a binary logic circuit, to convert a 3-bit binary number to it s 2 s complement. a 3-bit binary converter 2 s complement(a) New Example 55

53 Design a binary converter Design a binary logic circuit, to convert a 3-bit binary number to it s 2 s complement. How many inputs and outputs? Inputs = 3 Outputs =? Need to set up the truth table. 56

54 Design a binary converter Design a binary logic circuit, to convert a 3-bit binary number to it s 2 s complement. Binary ABC s complement XYZ 57

55 Design a binary converter Design a binary logic circuit, to convert a 3-bit binary number to it s 2 s complement. Binary 1 s complement 2 s complement ABC XYZ

56 Design a binary converter Design a binary logic circuit, to convert a 3-bit binary number to it s 2 s complement. Binary 1 s complement 2 s complement ABC XYZ XYZW The X=1 is like an overflow Write the output equations Simplify Implement 5 minutes 59

57 2 s Complement + Logic Expressions Decimal A B C X Y Z W

58 61

59 Solution 62

Odd-Prime Number Detector The table of minterms is represented. Table 13.1

Odd-Prime Number Detector The table of minterms is represented. Table 13.1 Minterm A B C D E 1 0 0 0 0 1 3 0 0 0 1 1 5 0 0 1 0 1 7 0 0 1 1 1 11 0 1 0 1 1 13 0 1 1 0 1 17 1 0 0 0 1 19 1 0 0 1 1 23 1 0 1