IS 151 Lecture 7
- 1. K-map Simplification of SOP Expressions • Steps for minimisation – Group the 1s – Determine the product term for each group – Sum the resulting product terms – A minimized SOP results into fewest possible terms with fewest possible variables IS 151 Digital Circuitry 1
- 2. K-map Simplification of SOP Expressions • Grouping the 1s – Group by enclosing adjacent cells containing 1s – The goal is to maximize the size of the groups and to minimize the number of groups – A group must contain either 1,2,4,8 or 16 cells – Each cell in a group must be adjacent to one or more cells in the same group, but all cells in the group do not have to be adjacent to each other – Always include the largest possible number of 1s in a group in accordance with the first rule above – 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 non-common 1s IS 151 Digital Circuitry 2
- 3. K-map Simplification of SOP Expressions • Examples C 0 1 C 0 1 00 1 1 1 01 1 1 11 AB 00 AB 1 01 11 1 10 A’B’C’ + BC + AB 10 1 1 1 A’C’ + AC + B’ IS 151 Digital Circuitry 3
- 4. K-map Simplification of SOP Expressions • Examples CD CD AB 00 01 00 1 1 01 1 1 11 11 10 1 1 AB’D + A’B + A’C’ 00 1 01 1 1 1 11 1 1 1 10 1 AB 00 1 10 01 11 10 1 1 1 1 AB’C + BC’ + D’ IS 151 Digital Circuitry 4
- 5. SOP Determination from the Map • Rules • Each group of cells containing 1s creates one product term composed of all variables that occur in only one form within the group. • Variables that occur complemented and uncomplemented are eliminated IS 151 Digital Circuitry 5
- 6. SOP Determination from the Map • For a 3-variable map – A 1-cell group yields a 3-variable product term (e.g. ABC) – A 2-cell group yields a 2-variable product term (e.g. AB) – A 4-cell group yields a 1-variable term (e.g. A) – An 8-cell group yields a value of 1 for the expression IS 151 Digital Circuitry 6
- 7. SOP Determination from the Map • For a 4-variable map – A 1-cell group yields a 4-variable product term (e.g. ABCD) – A 2-cell group yields a 3-variable product term (e.g. ABC) – A 4-cell group yields a 2-variable term (e.g. AB) – An 8-cell group yields a 1-variable term (e.g. A) – A 16-cell group yields a value of 1 for the expression IS 151 Digital Circuitry 7
- 8. SOP Determination from the Map • More examples CD AB 00 01 10 1 00 11 1 01 1 1 1 1 11 1 1 1 1 10 1 1 IS 151 Digital Circuitry 8
- 9. SOP Determination from the Map • Add a 1 at cell 1010 and determine the new product terms • Minimise the following expressions – AB’C + A’BC + A’B’C + A’B’C’ + AB’C’ – XY’Z + XYZ’ + X’YZ + X’YZ’ + XY’Z’ + XYZ – AB’C’D’ + A’B’C’D’ + A’BC’D’ + ABC’D’ + A’B’CD + AB’CD + A’B’CD’ + A’BCD’ + ABCD’ + AB’CD’ IS 151 Digital Circuitry 9
- 10. Mapping K-maps Directly from Truth Tables Inputs Output A B C X 0 0 0 0 0 0 1 1 01 0 1 0 0 11 C 1 1 00 0 0 0 1 1 0 0 1 1 1 1 0 1 1 1 1 1 0 1 1 AB 10 0 0 1 IS 151 Digital Circuitry 10
- 11. Don’t Care Conditions • Sometimes, some input variable combinations are not allowed • BCD – Binary Coded Decimal – A way to express each of the decimal digits (0-9) with a binary code of 4 bits • In BCD code (Chapter 2), there are 6 invalid combinations – 1010, 1011, 1100, 1101, 1110 and 1111 IS 151 Digital Circuitry 11
- 12. Don’t Care Conditions • Don’t care values BCD Codes 0000 1 CD 0 0001 AB 00 01 11 10 2 0010 00 0 1 3 2 3 0011 01 4 5 7 6 4 0100 11 12 13 15 14 5 0101 10 8 9 11 10 6 0110 7 0111 8 1000 9 1001 No BCD codes for 2digit decimal numbers (the don’t cares) IS 151 Digital Circuitry 12
- 13. Digital System Application • Introducing the first laboratory work – The Seven Segment Display (SSD) – Used in • Automobile instruments • Tablet counting control system, etc – Uses logic circuits that decode a binary coded decimal (BCD) number and activates the appropriate digits on the display IS 151 Digital Circuitry 13
- 14. The Seven Segment Display Seven segments: a, b, c, d, e, f and g E.g. check your digital watch, calculator, (some) phone display digits, etc IS 151 Digital Circuitry 14
- 15. Common SSDs • LED Displays – Consists of light-emitting diodes (LED) – Each segment is an LED that emits light when there is current through it • LCD Displays – Liquid Crystal Display – Operates by polarizing light so that a nonactivated segment reflects incident light and thus appears invisible against its background IS 151 Digital Circuitry 15
- 16. Segment Decoding Logic • Each segment is used for various decimal digits (i.e. 0 to 9) • Each segment must be activated by its own decoding circuit • The circuit detects the occurrence of any of the numbers in which the segment is used • Segments that are required to be activated for each digit (0-9) are: IS 151 Digital Circuitry 16
- 17. Segment Decoding Logic Digit Segments Activated 0 a, b, c, d, e, f 1 b, c 2 a, b, d, e, g 3 a, b, c, d, g 4 b, c, f, g 5 a, c, d, f, g 6 a, c, d, e, f, g 7 a, b, c 8 a, b, c, d, e, f, g 9 a, b, c, d, f, g IS 151 Digital Circuitry 17
- 18. Truth Table for the Segment Logic • Requires 4 BCD inputs and 7 outputs, one for each segment in the display (a to g) • Inputs: A, B, C and D – A – the least significant bit – D – the most significant bit – DCBA (as opposed to ABCD) IS 151 Digital Circuitry 18
- 19. Truth Table Decimal Digit Inputs Segment Outputs D C B A a b c d e f g 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 1 0 2 0 0 1 0 1 3 0 0 1 1 4 0 1 5 Digit Segments Activated 0 a, b, c, d, e, f 1 b, c 1 2 a, b, d, e, g 0 0 0 3 a, b, c, d, g 0 1 0 1 1 6 0 1 1 0 1 4 b, c, f, g 7 0 1 1 0 1 5 a, c, d, f, g 8 1 0 0 1 1 6 a, c, d, e, f, g 9 1 0 0 0 1 10 1 0 1 0 X 7 a, b, c 11 1 0 1 1 X 8 a, b, c, d, e, f, g 12 1 1 0 0 X 9 a, b, c, d, f, g 13 1 1 0 1 X 14 1 1 1 0 X 15 1 1 1 1 X IS 151 Digital Circuitry 19
- 20. Boolean Expressions for the Segment Logic • From the truth table, an SOP is developed for each of the segments • E.g. SOP for segment ‘a’ is – a = D’C’B’A’ + D’C’BA’ + D’C’BA + D’CB’A + D’CBA’ + D’CBA + DC’B’A’ + DC’B’A • Each product term represents each of the BCD inputs that activate that segment, therefore – implementing segment ‘a’ would require AND-OR circuit containing 8, 4-input AND gates and 1, 8-input OR gate – inverters are also required to produce the complement of each variable IS 151 Digital Circuitry 20
- 21. K-Map Minimisation for the Segment Logic • SOP for segment ‘a’ • Don’t care’s are used in forming BA groups DC 00 00 1 11 10 1 1 1 01 01 1 1 11 X X X X 10 1 1 X • a = B + D + CA + C’A’ X IS 151 Digital Circuitry 21
- 22. Laboratory Work • Obtain SOP for the remaining segments – b, c, d, e, f and g • For each segment, develop a K-Map and obtain a simplified expression for the segment • Draw a circuit for each segment • There will be duplicate terms across SOP expressions for the segments – identify them • For the final circuit, omit all but one duplicates, and draw the circuit using Deeds software in the laboratory • Prepare the laboratory report as instructed in the laboratory handout (report format) IS 151 Digital Circuitry 22
- 23. • End of lecture IS 151 Digital Circuitry 23