Gates – Part 2

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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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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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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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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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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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Converting English to Boolean

1. Identify phrases2. Identify connective words3. Construct a Boolean Expression4. Draw the Network

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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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AND/OR vs. OR/AND Logic Forms

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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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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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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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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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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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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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Types of Gates

• Gates already studied

– AND, NAND

– OR, NOR

– Inverter (NOT)

– XOR (Exclusive-OR)

– XNOR (Equivalence)

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NAND/NAND and NOR/NOR Logic

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AND/OR to NAND/NAND

AB+CD = [AB+CD]’’= [(AB)’(CD)’]’

Algebra-based:

Schematic-based:Preferred symbol in this context…

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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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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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Bubble Matching

How to make schematics readable, understandable,

maintainable, …

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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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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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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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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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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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Functional Completeness

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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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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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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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Functionally Complete

• Is the set {AND, OR} functionally complete?

• No! Can’t do a NOT

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How About NAND Only?

NOT

AND

OR

Success!

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How About NOR Only?

NOT

OR

AND

Success!

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Functional Completeness

• A few functionally complete sets:

AND, OR, NOT

OR, NOT

AND, NOT

NOR

NAND