IS 151 Lecture 7

IS 151 Digital Circuitry 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



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



Expressions• Examples

C

AB

0 1

00 1

01 1

11 1 1

10

C

AB

0 1

00 1 1

01 1

11 1

10 1 1

A’B’C’ + BC + AB A’C’ + AC + B’



Expressions• Examples

CD

AB 00 01 11 10

00 1 1

01 1 1 1 1

11

10 1 1

CD

AB 00 01 11 10

00 1 1

01 1 1 1

11 1 1 1

10 1 1 1

AB’D + A’B + A’C’ AB’C + BC’ + D’


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



• 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



• 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



• More examples

CD

AB 00 01 11 10

00 1 1

01 1 1 1 1

11 1 1 1 1

10 1 1



• 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’


Mapping K-maps Directly from

Truth TablesInputs Output

A B C X

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 0

1 0 0 1

1 0 1 0

1 1 0 0

1 1 1 1

C

AB

0 1

00 1

01

11 1

10 1


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


Don’t Care Conditions

• Don’t care valuesCD

AB 00 01 11 10

00 0 1 3 2

01 4 5 7 6

11 12 13 15 14

10 8 9 11 10

BCD Codes

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

No BCD codes for 2-

digit decimal numbers

(the don’t cares)


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


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


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 non-

activated segment reflects incident light and

thus appears invisible against its background


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:


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


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)


Truth TableDecimal

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 1

4 0 1 0 0 0

5 0 1 0 1 1

6 0 1 1 0 1

7 0 1 1 0 1

8 1 0 0 1 1

9 1 0 0 0 1

10 1 0 1 0 X

11 1 0 1 1 X

12 1 1 0 0 X

13 1 1 0 1 X

14 1 1 1 0 X

15 1 1 1 1 X

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


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


K-Map Minimisation for the

Segment Logic• SOP for segment ‘a’

BA

DC 00 01 11 10

00 1 1 1

01 1 1 1

11 X X X X

10 1 1 X X

• Don’t care’s are used in forming

groups

• a = B + D + CA + C’A’


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)

• End of lecture


IS 151 Lecture 7

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Transcript of IS 151 Lecture 7