Chapter 5 Karnaugh Maps
32
1 Chapter 5 Karnaugh Maps Chapter 5 Karnaugh Maps Mei Yang ECG 100-001 Logic Design 1 ECG 100-001 Logic Design 1
description
Chapter 5 Karnaugh Maps. ECG 100-001 Logic Design 1. Mei Yang. Minimum Forms of Switching Functions. When a function is realized using AND and OR gates, the cost is directly related the number of gates and gate inputs used. - PowerPoint PPT Presentation
Transcript of Chapter 5 Karnaugh Maps
1
Chapter 5 Karnaugh MapsChapter 5 Karnaugh Maps
Mei Yang
ECG 100-001 Logic Design 1ECG 100-001 Logic Design 1
2
Minimum Forms of Switching Minimum Forms of Switching FunctionsFunctions
When a function is realized using AND and OR gates, the cost is directly related the number of gates and gate inputs used.
Minimum SOP: A SOP form which (a) has a minimum number of terms and (b) for those expressions which have the same minimum number of terms, has a minimum number of literals.
Minimum POS: A POS form which (a) has a minimum number of terms and (b) for those expressions which have the same minimum number of terms, has a minimum number of literals.
3
Minimizing from Minterm Minimizing from Minterm formform
4
Why Kmap?Why Kmap?
Problems with the simplification method using Boolean algebra– The procedures are difficult to apply in
systematic way.– It is difficult to tell when you have arrived at
a minimum solution.Kmap is a systematic method, which is
especially useful for simplifying functions with three or four variables.
5
Plotting KmapsPlotting Kmaps
From truth tablesFrom minterm/maxterm expansionsFrom algebraic expressionsExamples on pp. 122-124
6
2-Variable Karnaugh Map2-Variable Karnaugh Map
7
3-Variable Karnaugh Map3-Variable Karnaugh MapBoolean adjacency -difference in one
variableGrouping adjacent 1’s
8
Grouping PrincipleGrouping Principle
Groupings can contain only 1s; no 0s.Groups can be formed only at right angles;
diagonal groups are not allowed.The number of 1s in a group must be a
power of 2 – even if it contains a single 1.The groups must be made as large as
possible.Groups can overlap and wrap around the
sides of the Kmap.Use the fewest number of groups possible.
9
Grouping of adjacent 1Grouping of adjacent 1’’ss
3-Variable Karnaugh Map3-Variable Karnaugh MapMultiple grouping
10
3-Variable Karnaugh Map3-Variable Karnaugh Map
Corresponding minterms
Adjacent minterms
11
3-Variable Karnaugh Map3-Variable Karnaugh Map
12
4-Variable Karnaugh Map4-Variable Karnaugh Map
13
4-Variable Karnaugh Map4-Variable Karnaugh Map
14
Boolean AdjacencyBoolean Adjacency
15
Grouping Grouping ExampleExample
16
Grouping ExampleGrouping Example
17
More than One Way of More than One Way of GroupingGrouping
18
A Systematic Way for A Systematic Way for MinimizationMinimization
Any single one or any group of ‘1’ s is called an IMPLICANTIMPLICANT of F.
A group (covering) that cannot be combined with some other 1’s or coverings to eliminate a variable is called PRIME PRIME IMPLICANTIMPLICANT..
19
Minimization with Don’t Minimization with Don’t CaresCares
Threat X’s as 1’s if you will get a larger grouping. Otherwise, treat them as 0’s.
20
Obtaining Minimum POS Obtaining Minimum POS Equation Equation
Grouping 0’s means obtaining a minimum SOP for F’.
21
Grouping ZerosGrouping Zeros
22
Minimization RulesMinimization Rules
The minimum SOP expression consists of some (but not necessarily all) of the prime implicants of a function.
If a SOP expression contains a term which is NOT a prime implicant, then it CANNOT be minimum.
23
Minimization RulesMinimization RulesHow to use prime implicants for obtaining a How to use prime implicants for obtaining a
minimum SOP equation?minimum SOP equation?
24
F(A,B,C,D)= BC’ +A’B’D
BC’ and A’B’D are called the Essential Prime ImplicantsEssential Prime Implicants, because they cover ‘1’s that cannot be covered by any other coverings.
A’CD is not an essential, and it is not included, because all 1’s are already covered, and there is no reason to add an extra term.
Essential Prime Essential Prime ImplicantsImplicants
25
More than one solutionMore than one solution
26
Two solutionsTwo solutions
27
Minimum SOPMinimum SOPTo obtain a minimum SOP equationInclude all essential prime implicants to the
equation.Check if all ones are covered by the
essential prime implicants.If there are remaining ‘1’s, include non-
essential prime implicants.There can be more than one minimum SOP
equation equally valid.
28
Minimization with Don’t Minimization with Don’t CaresCares
29
5-Variable Karnaugh Map5-Variable Karnaugh Map
30
5-Variable Karnaugh Map5-Variable Karnaugh Map
31
Example 1:Example 1:F = ∑m(1,2,3,5,6,8,9,10,11,13,24, 25, 26, 27,29)F = ∑m(1,2,3,5,6,8,9,10,11,13,24, 25, 26, 27,29)
32
Example 2: Example 2: F = ∑m (0,1,2,4,5,8,9,10,17,18,19,25,26)F = ∑m (0,1,2,4,5,8,9,10,17,18,19,25,26)
F = A'B'D'+C’D'E+C'DE’+
{AB’C’E or AB’C’D}+{A’BC’D’ or A’BC’E’}
DEBC
00 01 11 10
00
01
11
10
A=1/0
1
1
1
1
11
1
1
11
1
1
1
Essential prime Essential prime implicants are shown implicants are shown in sold line groups and in sold line groups and other prime implicants other prime implicants are shown in dotted are shown in dotted line groups. line groups.
