lec02 logic gates

8/2/2019 Lec02 Logic Gates

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CSSE1000 Lecture 2

Intro to Logic Gates

School of Information Technology and Electrical EngineeringThe University of Queensland


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Admin Issues

Circuit Schematics

Boolean Algebra

If you have questions, ask them at anyme

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Digital circuits

Only two logical levels present (i.e. binary)0 usually small voltage (e.g. around 0 volts)

1 usually larger voltage (e.g. 1.5 to 5 volts, dependingon the logic family, i.e. type/size of transistors)

Logic gates are the building blocks of computers;

Each gate has one or more inputs

perform operations (or functions) e.g.: NOT, AND, OR, NAND, NOR gates

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Inputs and outputs can have only two states, 1 and 0 Can be called true and false


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NOT gate inverts the signal

If A is 0 X is 1

A X

This is thes y m b o l for

If A is 1, X is 0 Gates o eration can be re resented usin a

a NOT gate

truth table:A X

1 0Input Output

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Output is 1 if all inputs are 1, i.e. ifin ut A a n d in ut B

a n d input Ca n d areall 1, t he output is 1 Logic symbol

Truth Table

A B X

2-input AND gate The opposite is a

A X 0 1 0

1 0 0

NAND gate (output

is 0 if all inputs are 1)

1 1 1

NANDA

X

0 0 1

0 1 1A

X

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B

1 1 0B

Bubble m eans inversion


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Output is 1 if at least one of the inputs is 1 i.e. ifin ut A o r in ut B

o r input Co r is 1,the output is 1

Logic symbolru a e

A B X

0 0 0OR

2-input OR gate The opposite is a

0 1 11 0 1

1 1 1B

X

NORgate (output

is 0 if any input is 1)NOR

AXX

ORA

0 0 1

0 1 0

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B

Bubble m eans inversion

B1 1 0


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For a 2-input gate Out ut is 1 ifexactl one of the in uts is 1

Lo ic s mbolTruth Table

A

BX

0 0 0

0 1 1

1 1 10This entry is the

difference between For > 2 inputs

Output is 1 if an odd number of inputs is 1

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Sometimes called the odd function


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Logic functions can be expressed

variables, e.g. A B X

unc ons, e.g. + .Rules about how this works called

Boolean algebra

take on values 0 or 1

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r v

Conventions well use: Inversion: overline

e.g. NOT(A) =

AND: dot(.) or implied (by adjacency)

e.g. AND(A,B) = AB = A.B OR: plus sign

e.g. OR(A,B,C) = A+B+C

Other examples:

NAND(A,B,C) = ABC

XOR(A,B) = A B = B + AB

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NOR(A,B) = A+B


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Summar of Lo ic FunctionRepresentations

There are four representations of logic

functions assume function ofn in uts Truth table

Lists output for all 2n combinations of inputs

Best to list inputs in a systematic way Boolean function (or equation)

output is 1

Lo ic Dia ram Combination of logic symbols joined by wires

Timing Diagram

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r v

Inputs are conventionally on left Outputs are

conventionally

A

on right

BM

ElectricalConnection

No Connection Lines represent electricalwires. Can be at hi h or

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low potential (logic 1 orlogic 0 respectively).


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Gates onIntegrated Circuits (ICs)

14 13 12 11 10 9

VCC

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n spac ng s .

x 0.3; chip is about1 2 3 4 5 6 7

GND

74LS00 has four 2-input NAND gates

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. . ,


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More than a logic diagram, a circuit schematic tells

you how to build the circuit

ICs are labelled U1, U2 etc (Other components also labelled)T e of IC iven also e. . 74LS00 or 74HCT00

Outputs labelled

Connectionswhich arentpart of the

Pin numbers are iven

separately

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Inputs are labelled


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Lo ic FunctionImplementation

Any logic function can be implemented as

the OR f AND c mbinati ns f the in uts Called sum of products

Consider truth table

A B C

0 0 0M

0

,write down the AND combination

of in uts that ive that 1

0 1 0

0 1 1

0

1

OR these together 1 0 1

1 1 0

1

1

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1 1 1 1


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Lo ic FunctionImplementation

A B C

0 0 0

M

00 0 1

0 1 0

0 1 1

0

0

1

1 0 0

1 0 1

1 1 0

0

1

1

1 1 1 1

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Exam le cont.Equivalent Logic Diagram

A

A

B C A B C

M = ABC + ABC + ABC + ABC

A

A B C4

A B C5

M8B

2

C

A B C

6

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A B C7C

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Sum of products does not necessarily produce circuit

with minimum number of gatesfunction Use rules of Boolean algebra (next slide)

Example: Z = AB + AC = A(B+C)

A AB

A A(B + C)

B

AB + AC

C

B

B + CC AC

22Fewer gates


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= + =

Name AND form OR form

Null law

0A = 0

1 + A = 1

Inverse law

= =

AA = 0 A + A = 1

Commutative law

Associative law

AB = BA

(AB)C = A(BC)

A + B = B + A

(A + B) + C = A + (B + C)

Distributative law

Absorption law

A + BC = (A + B)(A + C)

A(A + B) = A A + AB = A

A(B + C) = AB + AC

23De Morgan's law AB = A + B A + B = AB


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Express Z = A(B+C(A + B)) as a sum

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r w u v

AND/OR can be interchanged if you invert

the inputs and outputsAB = A + B A + B = AB

(a) (b)

A + BAB = AB=A + B

(c) (d)

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Homework: Use truth tables to convinceyourself that these are valid


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All circuits can be constructed from NAND or NOR gates

These are called complete gates Exam les:

NOT AND OR

A + BAB

AB

B

A A

A+

B

BA

A

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Reason: Easier to build NAND and NOR gates fromtransistors


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A B XOR

0 0 0

A

B

0 1 1

1 0 1 A

On any wire, you can introduce a

1 1 0

A

B

A

B

AA

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BB


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Go through the course profile

Do the reading if you havent already Do the homework (due at Prac sessions)

Sign-up (via session change request form) for

rac sess on

Tutorial session

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lec02 logic gates

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