Logic Gates

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Logic GatesLogic Gates

CS/APMA 202, Spring 2005CS/APMA 202, Spring 2005

Rosen, section 10.3Rosen, section 10.3

Aaron BloomfieldAaron Bloomfield

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Review of Boolean algebraReview of Boolean algebra

Just like Boolean logicJust like Boolean logic

Variables can only be 1 or 0Variables can only be 1 or 0 Instead of true / falseInstead of true / false

33


Not is a horizontal bar above the numberNot is a horizontal bar above the number 0 = 10 = 1 1 = 01 = 0

Or is a plusOr is a plus 0+0 = 00+0 = 0 0+1 = 10+1 = 1 1+0 = 11+0 = 1 1+1 = 11+1 = 1

And is multiplicationAnd is multiplication 0*0 = 00*0 = 0 0*1 = 00*1 = 0 1*0 = 01*0 = 0 1*1 = 11*1 = 1

__

__

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Example: translate (Example: translate (xx++yy++zz)()(xyzxyz) to a Boolean ) to a Boolean logic expressionlogic expression ((xxyyzz))((xxyyzz))

We can define a Boolean function:We can define a Boolean function: F(x,y) = (F(x,y) = (xxyy))((xxyy))

And then write a “truth table” for it:And then write a “truth table” for it:

__ __ __

xx yy F(x,y)F(x,y)

11 11 00

11 00 00

00 11 00

00 00 00

5

Quick surveyQuick survey

I understand the basics of Boolean I understand the basics of Boolean algebraalgebra

a)a) Absolutely!Absolutely!

b)b) More or lessMore or less

c)c) Not reallyNot really

d)d) Boolean what?Boolean what?

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Today’s demotivatorsToday’s demotivators

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Basic logic gatesBasic logic gates

NotNot

AndAnd

OrOr

NandNand

NorNor

XorXor

xx

xy

xy xy

xyz

zx+yx

yxy

x+y+z

z

xy

xy

x+yxy

xÅyxy

88

Rosen, §10.3 question 1Rosen, §10.3 question 1

Find the output of the following circuitFind the output of the following circuit

Answer: (Answer: (x+yx+y)y)y Or (Or (xxyy))yy

xy

x+y

y

(x+y)y

____

xy

y

99

x

y


Find the output of the following circuitFind the output of the following circuit

Answer: xyAnswer: xy Or Or ((xxyy) ≡ ) ≡ xxyy

x

y

x y x y

_ __ _ ______

10


I understand how to figure out I understand how to figure out what a logic gate doeswhat a logic gate does




d)d) Not at allNot at all

1111


Write the circuits for the following Write the circuits for the following Boolean algebraic expressionsBoolean algebraic expressions

a)a) xx++yy

xyxyxy

____

x x+y

1212

xyxyxy


Write the circuits for the following Write the circuits for the following Boolean algebraic expressionsBoolean algebraic expressions

b)b) ((xx++yy))xx______________

xy

x+yx+y (x+y)x

1313

Writing xor using and/or/notWriting xor using and/or/not

p p ÅÅ qq ( (pp qq) ) ¬( ¬(pp qq))

x x ÅÅ y y (x + y)( (x + y)(xyxy))

xx yy xxÅÅyy

11 11 00

11 00 11

00 11 11

00 00 00

xy

x+y

xy xy

(x+y)(xy)

________

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I understand how to write a logic I understand how to write a logic circuit for simple Boolean formulacircuit for simple Boolean formula





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Converting decimal numbers to Converting decimal numbers to binarybinary

53 53 = 32 + 16 + 4 + 1= 32 + 16 + 4 + 1= 2= 255 + 2 + 244 + 2 + 222 + 2 + 200

= 1*2= 1*255 + 1*2 + 1*244 + 0*2 + 0*233 + 1*2 + 1*222 + 0*2 + 0*211 + 1*2 + 1*200

= 110101 in binary= 110101 in binary= 00110101 as a full byte in binary= 00110101 as a full byte in binary

211= 128 + 64 + 16 + 2 + 1211= 128 + 64 + 16 + 2 + 1= 2= 277 + 2 + 266 + 2 + 244 + 2 + 211 + 2 + 200

= 1*2= 1*277 + 1*2 + 1*266 + 0*2 + 0*255 + 1*2 + 1*244 + 0*2 + 0*233 + 0*2 + 0*222 + + 1*21*211 + 1*2 + 1*200

= 11010011 in binary= 11010011 in binary

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Converting binary numbers to Converting binary numbers to decimaldecimal

What is 10011010 in decimal?What is 10011010 in decimal?1001101010011010 = 1*2 = 1*277 + 0*2 + 0*266 + 0*2 + 0*255 + 1*2 + 1*244 + 1*2 + 1*233 + +

0*20*222 + 1*2 + 1*211 + 0*2 + 0*200

= 2= 277 + 2 + 244 + 2 + 233 + 2 + 211

= 128 + 16 + 8 + 2= 128 + 16 + 8 + 2 = 154= 154

What is 00101001 in decimal?What is 00101001 in decimal? 00101001 = 0*200101001 = 0*277 + 0*2 + 0*266 + 1*2 + 1*255 + 0*2 + 0*244 + 1*2 + 1*233 + +

0*20*222 + 0*2 + 0*211 + 1*2 + 1*200

= 2= 255 + 2 + 233 + 2 + 200

= 32 + 8 + 1= 32 + 8 + 1 = 41= 41

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A bit of binary humorA bit of binary humor

Available for $15 at Available for $15 at http://www.thinkgeek.cohttp://www.thinkgeek.com/ m/ tshirts/frustrations/5aa9tshirts/frustrations/5aa9//

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I understand the basics of I understand the basics of converting numbers between converting numbers between decimal and binarydecimal and binary





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How to add binary numbersHow to add binary numbers

Consider adding two 1-bit binary numbers Consider adding two 1-bit binary numbers xx and and yy 0+0 = 00+0 = 0 0+1 = 10+1 = 1 1+0 = 11+0 = 1 1+1 = 101+1 = 10

Carry is Carry is xx AND AND yy

Sum is Sum is xx XOR XOR yy

The circuit to compute this is called a half-adderThe circuit to compute this is called a half-adder

xx yy CarryCarry SumSum

00 00 00 00

00 11 00 11

11 00 00 11

11 11 11 00

2020

The half-adderThe half-adder

Sum = Sum = xx XOR XOR yy

Carry = Carry = xx AND AND yy

xy Sum

Carry

xy Sum

Carry

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Using half addersUsing half adders

We can then use a half-adder to compute We can then use a half-adder to compute the sum of two Boolean numbersthe sum of two Boolean numbers

1 1 0 01 1 0 0

+ 1 1 1 0+ 1 1 1 0001100??

000011

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I understand half addersI understand half adders





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How to fix thisHow to fix this

We need to create an adder that can take a We need to create an adder that can take a carry bit as an additional inputcarry bit as an additional input Inputs: Inputs: xx, , yy, carry in, carry in Outputs: sum, carry outOutputs: sum, carry out

This is called a full adderThis is called a full adder Will add Will add xx and and yy with a half-adder with a half-adder Will add the sum of that to the Will add the sum of that to the

carry incarry in

What about the carry out?What about the carry out? It’s 1 if either (or both):It’s 1 if either (or both): xx++yy = 10 = 10 xx++yy = 01 and carry in = 1 = 01 and carry in = 1

xx yy cc carrycarry sumsum

11 11 11 11 11

11 11 00 11 00

11 00 11 11 00

11 00 00 00 11

00 11 11 11 00

00 11 00 00 11

00 00 11 00 11

00 00 00 00 00

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The full adderThe full adder

The “HA” boxes are half-addersThe “HA” boxes are half-adders

HAX

Y

S

C

HAX

Y

S

C

xy

c

c

s

HAX

Y

S

C

HAX

Y

S

C

xy

c

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The full adderThe full adder

The full circuitry of the full adderThe full circuitry of the full adder

xy

s

c

c

2626

Adding bigger binary numbersAdding bigger binary numbers

Just chain full adders togetherJust chain full adders together

HAX

Y

S

C

FAC

Y

X

S

C

FAC

Y

X

S

C

FAC

Y

X

S

C

x1y1

x2y2

x3y3

x0y0

s0

s1

s2

s3c

2727

Adding bigger binary numbersAdding bigger binary numbers

A half adder has 4 logic gatesA half adder has 4 logic gatesA full adder has two half adders plus a OR gateA full adder has two half adders plus a OR gate Total of 9 logic gatesTotal of 9 logic gates

To add To add nn bit binary numbers, you need 1 HA and bit binary numbers, you need 1 HA and nn-1 FAs-1 FAsTo add 32 bit binary numbers, you need 1 HA To add 32 bit binary numbers, you need 1 HA and 31 FAsand 31 FAs Total of 4+9*31 = 283 logic gatesTotal of 4+9*31 = 283 logic gates

To add 64 bit binary numbers, you need 1 HA To add 64 bit binary numbers, you need 1 HA and 63 FAsand 63 FAs Total of 4+9*63 = 571 logic gatesTotal of 4+9*63 = 571 logic gates

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I understand (more or less) about I understand (more or less) about adding binary numbers using logic adding binary numbers using logic gatesgates





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More about logic gatesMore about logic gates

To implement a logic gate in hardware, To implement a logic gate in hardware, you use a transistoryou use a transistor

Transistors are all enclosed in an “IC”, or Transistors are all enclosed in an “IC”, or integrated circuitintegrated circuit

The current Intel Pentium IV processors The current Intel Pentium IV processors have 55 million transistors!have 55 million transistors!

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Pentium math error 1Pentium math error 1 Intel’s Pentiums Intel’s Pentiums

(60Mhz – 100 Mhz) (60Mhz – 100 Mhz) had a floating point had a floating point errorerror

Graph of z = y/xGraph of z = y/x

Intel reluctantlyIntel reluctantlyagreed to replace agreed to replace them in 1994them in 1994

Graph from http://kuhttp.cc.ukans.edu/cwis/units/IPPBR/pentium_fdiv/pentgrph.htmlGraph from http://kuhttp.cc.ukans.edu/cwis/units/IPPBR/pentium_fdiv/pentgrph.html

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Pentium math error 2Pentium math error 2 Top 10 reasons to buy a Pentium:Top 10 reasons to buy a Pentium:

1010 Your old PC is too accurate Your old PC is too accurate

8.9999163362 8.9999163362 Provides a good alibi when the IRS calls Provides a good alibi when the IRS calls

7.9999414610 7.9999414610 Attracted by Intel's new "You don't need to know what's Attracted by Intel's new "You don't need to know what's

inside" campaign inside" campaign

6.9999831538 6.9999831538 It redefines computing--and mathematics!It redefines computing--and mathematics!

5.9999835137 5.9999835137 You've always wondered what it would be like to be a You've always wondered what it would be like to be a

plaintiff plaintiff

4.9999999021 4.9999999021 Current paperweight not big enough Current paperweight not big enough

3.9998245917 3.9998245917 Takes concept of "floating point" to a new level Takes concept of "floating point" to a new level

2.9991523619 2.9991523619 You always round off to the nearest hundred anyway You always round off to the nearest hundred anyway

1.9999103517 1.9999103517 Got a great deal from the Jet Propulsion Laboratory Got a great deal from the Jet Propulsion Laboratory

0.9999999998 0.9999999998 It'll probably work!! It'll probably work!!

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Flip-flopsFlip-flops

Consider the following circuit:Consider the following circuit:

What does it do?What does it do?

3333

MemoryMemory

A flip-flop holds a single bit of memoryA flip-flop holds a single bit of memory The bit “flip-flops” between the two NAND The bit “flip-flops” between the two NAND

gatesgates

In reality, flip-flops are a bit more In reality, flip-flops are a bit more complicatedcomplicated

Have 5 (or so) logic gates (transistors) per flip-Have 5 (or so) logic gates (transistors) per flip-flopflop

Consider a 1 Gb memory chipConsider a 1 Gb memory chip 1 Gb = 8,589,934,592 bits of memory1 Gb = 8,589,934,592 bits of memory That’s about 43 million transistors!That’s about 43 million transistors!

In reality, those transistors are split into 9 In reality, those transistors are split into 9 ICs of about 5 million transistors eachICs of about 5 million transistors each

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I felt I understood the material in this I felt I understood the material in this slide set…slide set…

a)a) Very wellVery well

b)b) With some review, I’ll be goodWith some review, I’ll be good



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The pace of the lecture for this The pace of the lecture for this slide set was…slide set was…

a)a) FastFast

b)b) About rightAbout right

c)c) A little slowA little slow

d)d) Too slowToo slow

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Logic Gates