06 GATES2 Page 1 ECEn 224 © 2003-2008 BYU Gates – Part 2
Dec 30, 2015
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ECEn 224 © 2003-2008BYU
Gates – Part 2
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ECEn 224 © 2003-2008BYU
Converting English to Boolean Expressions
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ECEn 224 © 2003-2008BYU
The air conditioner should be turned on if and only if: - the temperature is greater than 75, - the time is between 8a.m. and 5 p.m., - and it is not a holiday.
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ECEn 224 © 2003-2008BYU
1. Identify phrases
F = air conditioner should be turned onA = temperature is greater than 75B = time is between 8a.m. and 5 p.mC = it is a holiday
The air conditioner should be turned on if and only if: - the temperature is greater than 75, - the time is between 8a.m. and 5 p.m., - and it is not a holiday.
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ECEn 224 © 2003-2008BYU
2. Identify connective words =
The air conditioner should be turned on if and only if: - the temperature is greater than 75, - the time is between 8a.m. and 5 p.m., - and it is not a holiday.
AND
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ECEn 224 © 2003-2008BYU
3. Construct a Boolean Expression
F = air conditioner should be turned onA = temperature is greater than 75B = time is between 8a.m. and 5 p.mC = it is a holiday
F = A • B • C’
The air conditioner should be turned on if and only if: - the temperature is greater than 75, - the time is between 8a.m. and 5 p.m., - and it is not a holiday.
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ECEn 224 © 2003-2008BYU
4. Draw the Network
F = A • B • C’
The air conditioner should be turned on if and only if: - the temperature is greater than 75, - the time is between 8a.m. and 5 p.m., - and it is not a holiday.
BA
CF
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ECEn 224 © 2003-2008BYU
Converting English to Boolean
1. Identify phrases2. Identify connective words3. Construct a Boolean Expression4. Draw the Network
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ECEn 224 © 2003-2008BYU
Converting English to Boolean
• Be careful: Boolean algebra is precise, English is not.
The roads will be very slippery if it snows or rains and there is oil on the road.
F = A + BC
or
F = (A + B) C
AB C
Which is it?
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ECEn 224 © 2003-2008BYU
AND/OR vs. OR/AND Logic Forms
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ECEn 224 © 2003-2008BYU
AND/OR Logic from Truth Table
1. Write the SOP by inspection for f
A
B
C
f
f = A’B’C +AB’C + ABC’ + ABC
A B C f
0 0 0 00 0 1 10 1 0 00 1 1 01 0 0 01 0 1 11 1 0 11 1 1 1
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ECEn 224 © 2003-2008BYU
AND/OR Logic from Truth Table
2. Simplify the equation
f = A’B’C +AB’C + ABC’ + ABC
f = (A + A’) B’C + AB (C + C’)
f = AB + B’C
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ECEn 224 © 2003-2008BYU
AND/OR Logic from Truth Table
3. Draw the logic network
f = AB + B’C
A
B
C
f
“AND/OR”
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ECEn 224 © 2003-2008BYU
OR/AND Logic from Truth Table
A
B
C
f
f ’ = A’B’C’ + A’BC’ + AB’C’f = (A’B’C’ + A’BC’ + AB’C’)’f = (A + B + C)(A + B’ + C)(A’ + B + C)
1. Write a POS by inspection from f
A B C f
0 0 0 00 0 1 10 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 1
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ECEn 224 © 2003-2008BYU
OR/AND Logic from Truth Table
3. Simplify the equation:
f = (A + B + C)(A + B’ + C)(A’ + B + C)f = (A + B)(A + B’)(A’ + B) + Cf = (A)(A’ + B) + Cf = (AA’ + AB) + Cf = AB + Cf = (A + C)(B + C)
A + BCD = A + B(CD)= (A + B)(A + CD)= (A + B)(A + C)(A + D)
(X + Y)(X + Y’) = X
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ECEn 224 © 2003-2008BYU
OR/AND Logic from Truth Table
4. Draw the logic network
f = (A+C)(B+C)
BC
AC
f
“OR/AND”
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ECEn 224 © 2003-2008BYU
Types of Gates
• Gates already studied
– AND, NAND
– OR, NOR
– Inverter (NOT)
– XOR (Exclusive-OR)
– XNOR (Equivalence)
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ECEn 224 © 2003-2008BYU
NAND/NAND and NOR/NOR Logic
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ECEn 224 © 2003-2008BYU
AND/OR to NAND/NAND
AB+CD = [AB+CD]’’= [(AB)’(CD)’]’
Algebra-based:
Schematic-based:Preferred symbol in this context…
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ECEn 224 © 2003-2008BYU
OR/AND to NOR/NOR
(A+B)(C+D) = [(A+B)(C+D)]’’= [(A+B)’ + (C+D)’]’
Algebra-based:
Schematic-based:Preferred symbol in this context…
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ECEn 224 © 2003-2008BYU
Alternative Gate SymbolsWhich is easier to understand?
A
B
C
D
Q = AB + CD
A
B
C
D
Q = ?
A
B
C
D
Q = (A+B)(C+D)
A
B
C
D
Q = ?
If you think of the bubbles as canceling each other out…
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ECEn 224 © 2003-2008BYU
Bubble Matching
How to make schematics readable, understandable,
maintainable, …
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ECEn 224 © 2003-2008BYU
Bubble Matching Rules
• Choose alternative symbols• Match all interior bubbles • More than one solution• Makes reading of the function trivial
AB
CD
FAB
CD
F
F = [(AB)’(C+D)’]’ ??? F = AB + (C+D) = AB+C+D
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ECEn 224 © 2003-2008BYU
More Bubble Matching
AB
CD
FAB
CD
F
This doesn’t work – has unmatched bubbles
AB
CD
F
This works. F = AB + C’D’
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ECEn 224 © 2003-2008BYU
Yet More Bubble Matching
AB
CD
F
Alternative solution = convert top-left gate.
F’ = (A’+B’)(C+D)F = AB+C’D’
Same result as on previous slide
AB
CD
F
Same circuit as on previous slide…
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ECEn 224 © 2003-2008BYU
Can Bubbles Always Be Matched?
• No…
A
B
CD
EF
x
y
Nodes x and y both drive final gate and so both need same polarity (bubble or no bubble).
Can’t satisfy that requirement because x also drives y’s input.
This is called reconvergent fanout
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ECEn 224 © 2003-2008BYU
Bubble Matching Summary
• Convert symbols to match bubbles– Two versions for each circuit
• Inverted output• Non-inverted output
• Good schematic style similar to good programming style– Convey meaning as well as function– Document the design
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ECEn 224 © 2003-2008BYU
Functional Completeness
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ECEn 224 © 2003-2008BYU
Functionally Complete
• AND, OR, and NOT make a functionally complete set of operators– There is no truth table which cannot be
implemented using AND, OR, NOT– Any set of gates which can implement AND, OR
and NOT is also functionally complete
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ECEn 224 © 2003-2008BYU
Functionally Complete
• Is the set {AND, NOT} functionally complete?• If I could just build an OR gate…
or…
X + Y = (X’Y’)’
Success!
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ECEn 224 © 2003-2008BYU
Functionally Complete
• Is the set {OR, NOT} functionally complete?• If I could just build an AND gate…
Success!
or…
XY = (X’ + Y’)’
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ECEn 224 © 2003-2008BYU
Functionally Complete
• Is the set {AND, OR} functionally complete?
• No! Can’t do a NOT
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ECEn 224 © 2003-2008BYU
How About NAND Only?
NOT
AND
OR
Success!
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ECEn 224 © 2003-2008BYU
How About NOR Only?
NOT
OR
AND
Success!
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ECEn 224 © 2003-2008BYU
Functional Completeness
• A few functionally complete sets:
AND, OR, NOT
OR, NOT
AND, NOT
NOR
NAND