Post on 11-Aug-2020
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ENGR 303 – Introduction to Logic Design Lecture 4
Dr. Chuck BrownEngineering and Computer Information Science
Folsom Lake College
<2>
• Logic to Gates
• Karnaugh Maps (K-Maps)
Outline for Todays Lecture
ENGR 303
<3>
• Two-level logic: ANDs followed by ORs
• Example: Y = ABC + ABC + ABC
BA C
Y
minterm: ABC
minterm: ABC
minterm: ABC
A B C
From Logic to Gates
ENGR 303
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• Inputs on the left (or top)
• Outputs on right (or bottom)
• Gates flow from left to right
• Straight wires are best
Circuit Schematics Rules
ENGR 303
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A1
A0
0 00 11 01 1
Y3Y2
Y1
Y0A
3A2
0 00 00 00 0
0 00 10 11 01 10 0
0 10 10 11 0
0 11 01 01 10 00 1
1 01 01 11 1
1 01 11 11 1
A0
A1
PRIORITY
CiIRCUIT
A2
A3
Y0
Y1
Y2
Y3
• Example: Priority Circuit
Output asserted
corresponding to
most significant
TRUE input
Multiple-Output Circuits
ENGR 303
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0
A1
A0
0 00 11 01 1
0
00
Y3Y2
Y1
Y0
0000
0011
0100
A3
A2
0 00 00 00 0
0 0 0 1 0 00 10 11 01 10 0
0 10 10 11 0
0 11 01 01 10 00 1
1 01 01 11 1
1 01 11 11 1
0001
1110
0000
0000
1 0 0 01111
0000
0000
0000
1 0 0 01 0 0 0
A0
A1
PRIORITY
CiIRCUIT
A2
A3
Y0
Y1
Y2
Y3
• Example: Priority Circuit
Output asserted
corresponding to
most significant
TRUE input
Multiple-Output Circuits
ENGR 303
<7>
A1
A0
0 00 11 01 1
0000
Y3Y2
Y1
Y0
0000
0011
0100
A3
A2
0 00 00 00 0
0 0 0 1 0 00 10 11 01 10 0
0 10 10 11 0
0 11 01 01 10 00 1
1 01 01 11 1
1 01 11 11 1
0001
1110
0000
0000
1 0 0 01111
0000
0000
0000
1 0 0 01 0 0 0
A3A
2A
1A
0
Y3
Y2
Y1
Y0
Don’t Care Simplification
ENGR 303
A1
A0
0 00 11 XX X
0000
Y3Y2
Y1
Y0
0001
0010
0100
A3
A2
0 00 00 00 1
X X 1 0 0 01 X
But is there an easier way?
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A1
A0
0 00 11 01 1
0000
Y3Y2
Y1
Y0
0000
0011
0100
A3
A2
0 00 00 00 0
0 0 0 1 0 00 10 11 01 10 0
0 10 10 11 0
0 11 01 01 10 00 1
1 01 01 11 1
1 01 11 11 1
0001
1110
0000
0000
1 0 0 01111
0000
0000
0000
1 0 0 01 0 0 0
A1
A0
0 00 11 XX X
0000
Y3Y2
Y1
Y0
0001
0010
0100
A3
A2
0 00 00 00 1
X X 1 0 0 01 X
Don’t Cares
ENGR 303
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• Boolean expressions can be minimized by combining terms
• E.g. Y=A’B’C’ + A’B’C -> A’B’(C’+C) -> Y=A’B’
• K-maps minimize equations graphically
Karnaugh Maps (K-Maps)
ENGR 303
A B C Y
0 0 0 1
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 0
A’B’C’ A’B’C A’BC A’BC’
AB’C’ AB’C ABC ABC’
A B’C’ B’C BC BC’
A’
A
BC
1 1 0 0
0 0 0 0
A 00 01 11 10
0
1
BC
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• Circle 1’s in adjacent squares
• In Boolean expression, include only
literals whose true and complement form
are not in the circle
Y = A’B’
B C0 0
0 1
1 0
1 1
A0
0
0
0
0 0
0 1
1 0
1 1
1
1
1
1
1
1
0
0
0
0
0
0
Y
K-Map
ENGR 303
1 1 0 0
0 0 0 0
A 00 01 11 10
0
1
BC
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• Complement: variable with a bar over it
A, B, C
• Literal: variable or its complement
A, A, B, B, C, C
• Implicant: product of literals
ABC, AC, BC
• Prime implicant: implicant corresponding to the largest circle in a K-map
K-Map Definitions
ENGR 303
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• Every 1 must be circled at least once
• Each circle must span a power of 2 (i.e. 1, 2, 4) squares in each direction
• Each circle must be as large as possible
• A circle may wrap around the edges
• A “don't care” (X) is circled only if it helps minimize the equation
K-Map Rules
ENGR 303
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3-Input K-Map
ENGR 303
A B C Y
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
0 0 1 1
0 0 1 0
A 00 01 11 10
0
1
BC
A’B’C’ A’B’C A’BC A’BC’
AB’C’ AB’C ABC ABC’
A B’C’ B’C BC BC’
A’
A
BC
Y = A’B + BC
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0
C D0 0
0 1
1 0
1 1
B0
0
0
0
0 0
0 1
1 0
1 1
1
1
1
1
1
1
1
0
1
1
1
YA0
0
0
0
0
0
0
0
0 0
0 1
1 0
1 1
0
0
0
0
0 0
0 1
1 0
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
4-Input K-Map
ENGR 303
00
1 0 1 1
0 1 1 1
0 0 0 0
1 1 0 1
ABCD
01 11 10
00
01
11
10
Y = A’C + A’BD + AB’C’ + B’D’
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0
C D0 0
0 1
1 0
1 1
B0
0
0
0
0 0
0 1
1 0
1 1
1
1
1
1
1
1
1
0
X
1
1
YA0
0
0
0
0
0
0
0
0 0
0 1
1 0
1 1
0
0
0
0
0 0
0 1
1 0
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
X
X
X
X
X
X
K-Maps with Don’t Cares
ENGR 303
00
1 0 1 1
0 X 1 1
X X X X
1 1 X X
ABCD
01 11 10
00
01
11
10
Y = A + B’D’ + C