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Digital Logic & Design Vishal Jethava Lecture 12 svbitec.wordpress.com svbitec.wordpress.com
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Digital Logic & Design
Vishal Jethava
Lecture 12
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Recap
Karnaugh Maps Mapping Standard POS expressions Mapping Non-Standard POS expressions Simplification of K-maps for POS
expressions SOP-POS conversion using K-map 5-variable K-map Functions having multiple outputs
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Comparator Circuit
Inputs two 2-bit binary numbers A and B Has three outputs A>B A=B A<B
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Inputs Output
A1 A0 B1 B0 A>B
0 0 0 0 0
0 0 0 1 0
0 0 1 0 0
0 0 1 1 0
0 1 0 0 1
0 1 0 1 0
0 1 1 0 0
0 1 1 1 0
Inputs Output
A1 A0 B1 B0 A>B
1 0 0 0 1
1 0 0 1 1
1 0 1 0 0
1 0 1 1 0
1 1 0 0 1
1 1 0 1 1
1 1 1 0 1
1 1 1 1 0
Function Table for A>B
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Inputs Output
A1 A0 B1 B0 A=B
0 0 0 0 1
0 0 0 1 0
0 0 1 0 0
0 0 1 1 0
0 1 0 0 0
0 1 0 1 1
0 1 1 0 0
0 1 1 1 0
Inputs Output
A1 A0 B1 B0 A=B
1 0 0 0 0
1 0 0 1 0
1 0 1 0 1
1 0 1 1 0
1 1 0 0 0
1 1 0 1 0
1 1 1 0 0
1 1 1 1 1
Function Table for A=B
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Inputs Output
A1 A0 B1 B0 A<B
0 0 0 0 0
0 0 0 1 1
0 0 1 0 1
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 1
0 1 1 1 1
Inputs Output
A1 A0 B1 B0 A<B
1 0 0 0 0
1 0 0 1 0
1 0 1 0 0
1 0 1 1 1
1 1 0 0 0
1 1 0 1 0
1 1 1 0 0
1 1 1 1 0
Function Table for A<B
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Karnaugh Map for A>B
A1A0/B1B0
00 01 11 10
00 0 0 0 0
01 1 0 0 0
11 1 1 0 1
10 1 1 0 0
00101011 BAABBABA
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Karnaugh Map for A=B
A1A0/B1B0
00 01 11 10
00 1 0 0 0
01 0 1 0 0
11 0 0 1 0
10 0 0 0 1
0101010101010101 BBAABBAABBAABBAA
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Karnaugh Map for A<B
A1A0/B1B0
00 01 11 10
00 0 1 1 1
01 0 0 1 1
11 0 0 0 0
10 0 0 1 0
01000111 BBABAABA
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Quine-McCluskey Method
Difficult to manage K-maps of more than 4 variables
With a 4-varaible K-map optimum groups of 1s and 0s are not formed
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Karnaugh Map
AB/CD
00 01 11 10
00 0 1 1 0
01 0 0 1 1
11 1 1 1 1
10 1 1 1 0
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Karnaugh Map
AB/CD
00 01 11 10
00 0 1 0 0
01 0 1 1 1
11 1 1 1 0
10 0 0 1 0
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Quine-McCluskey Method
Programmed based approach Two step method Find Prime Implicants through exhaustive
search Selecting minimal set of essential prime
implicants
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Quine-McCluskey Method (table1)
Minterm A B C D
1 0 0 0 1
3 0 0 1 1
6 0 1 1 0
7 0 1 1 1
8 1 0 0 0
9 1 0 0 1
11 1 0 1 1
12 1 1 0 0
13 1 1 0 1
14 1 1 1 0
15 1 1 1 1svbitec.wordpress.comsvbitec.wordpress.com
Quine-McCluskey Method (table2)
Minterm A B C D used
1 0 0 0 1 8 1 0 0 0 3 0 0 1 1 6 0 1 1 0 9 1 0 0 1 12 1 1 0 0 7 0 1 1 1 11 1 0 1 1 13 1 1 0 1 14 1 1 1 0 15 1 1 1 1 svbitec.wordpress.comsvbitec.wordpress.com
Quine-McCluskey Method (table3)
A B C D used
1,3 0 0 - 1 1,9 - 0 0 1 8,9 1 0 0 - 8,12 1 - 0 0 3,7 0 - 1 1 3,11 - 0 1 1 6,7 0 1 1 - 6,14 - 1 1 0 9,11 1 0 - 1 9,13 1 - 0 1
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Quine-McCluskey Method (table3)
A B C D used
12,13 1 1 0 - 12,14 1 1 - 0 7,15 - 1 1 1 11,15 1 - 1 1 13,15 1 1 - 1 14,15 1 1 1 -
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Quine-McCluskey Method (table4)
A B C D used
1,3,9,11 - 0 - 1
8,9,12,13 1 - 0 -
3,7,11,15 - - 1 1
6,7,14,15 - 1 1 -
9,11,13,15 1 - - 1
12,13,14,15 1 1 - -
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Quine-McCluskey Method (table5)
DB
CA
CD
BC
AD
AB
1 3 6 7 8 9 11 12 13 14 15
x x x x
x x x x
x x x x
x x x x
x x x x
x x x x
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Example 2
Slightly different method that uses binary values
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Quine-McCluskey Method (table1)
Minterm A B C D
1 0 0 0 1
5 0 1 0 1
6 0 1 1 0
7 0 1 1 1
11 1 0 1 1
12 1 1 0 0
13 1 1 0 1
15 1 1 1 1
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Quine-McCluskey Method (table2)
Minterm A B C D Used
1 0 0 0 1
5 0 1 0 1
6 0 1 1 0
12 1 1 0 0
7 0 1 1 1
11 1 0 1 1
13 1 1 0 1
15 1 1 1 1
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Quine-McCluskey Method (table3)
Minterms Variable removed used
1,5 4
5,7 2 5,13 8 6,7 1
12,13 1
7,15 8 11,15 4
13,15 2 svbitec.wordpress.comsvbitec.wordpress.com
Quine-McCluskey Method (table4)
Minterms Term removed used
5,7,13,15 2,8
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Quine-McCluskey Method (table5)
DCA
BCA
CAB
ACD
BD
1 5 6 7 11 12 13 15
x x
x x
x x
x x
x x x x
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Comparator that compares two 3-bit numbers
6 variables, 64 input combinations Representing the comparator function
through function table long and tedious Represent three output functions in terms
of minterms Solve by Quine-McClusky method
Quine-McCluskey Method
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Odd Prime Number Circuit
Circuit detects odd prime numbers for a 5-bit input number
Function represented by minterms
1, 3, 5, 7, 11, 13, 17, 19, 23, 29 and 31
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