1 Basic Digital Logic. Prof Jess Role @UEAB 2008 ◊Digital Electronics ◊Number Systems and Logic ◊Electronic Gates ◊Combinational Logic ◊Sequential Circuits

1

Basic Digital Logic

Prof Jess Role @UEAB 2008Prof Jess Role @UEAB 2008

◊ Digital Electronics ◊ Number Systems and Logic◊ Electronic Gates◊ Combinational Logic◊ Sequential Circuits◊ ADC – DAC circuits◊ Memory and Microprocessors◊ Hardware Description Languages


Weekly Structure

◊ Lectures Monday, Tuesday, Wednesday

◊ Slides in ppt and pdf format on support website:◊ http://www.roletech.net/comp360.htm

(follow link from course website)

◊ Tutorials anytime - Sample Questions on website.


The lecture today

◊ Digital vs Analog data◊ Binary inputs and outputs◊ Binary, octal, decimal and

hexadecimal number systems◊ Other uses of binary coding.


Analog/Analogue Systems

◊ Analogue Systems◊ V(t) can have any value between its

minimum and maximum value

V(t)


Digital Systems◊ Digital Systems

◊ V(t) must take a value selected from a set of values called an alphabet

◊ Binary digital systems form the basis of almost all hardware systems currently

V(t)

For example, Binary Alphabet: 0, 1.

1 0 1 0 1


Slide example

◊ Consider a child’s slide in a playground:

continuous movement

a set of discrete steps

levels


Relationship between Analogue and Digital systems◊ Advantages of Digital Systems

◊ Analogue systems: slight error in input yields large error in output

◊ Digital systems more accurate and reliable

◊ Computers use digital circuits internally

◊ Interface circuits (for instance, sensors and actuators) are often analogue

5 Volt

0 Volt

0.80.4

2.42.8

InputRangefor 1

InputRangefor 0

OutputRangefor 0

OutputRangefor 1


Exercise

◊ Explain whether the following are analog or digital:

◊ A photograph or painting◊ A scanned image◊ Sound from a computer’s loud speaker◊ Sound file stored on disc

Prof Jess Role @UEAB 2008

Binary Inputs and Outputs

◊ Coding:◊ A single binary input can

only have two values: True or False (Yes or No) (1 or 0)


Binary◊ More bits = more combinations

0 0 0 1 1 0 1 1

Each additional input doubles the number of combinations we can representi.e. with n inputs it is possible to represent 2n combinations


Combinations

◊ Example 1:◊ How many combinations are possible

with 10 binary inputs?

◊ Example 2:◊ What is the minimum number of bits

needed to represent the digits ‘0’ to ‘9’ as a binary code?”


Decimal systems

◊ Number Representation◊ Difficult to represent Decimal numbers

directly in a digital system◊ Easier to convert them to binary◊ There is a weighting system:

eg 403 = 4 x 100 + 0 x 10 + 3 x 1

or in, powers of 10:

40310= 4x102 + 0x101 + 3x100 = 400 + 0 + 3



◊ Both Decimal and Binary numbers use a positional weighting system, eg:

10102 = 1x23+0x22+1x21+0x20 = 1x8 + 0x4 + 1x2 + 0x1 = 1010

decimal 100 (102) 10 (101) 1 (100)

4 0 3 400 + 0 + 3

binary 8 (23) 4 (22) 2 (21) 1 (20)

1 0 0 1 8 + 0 + 0 + 1


Binary to decimal◊ Multiply each 1 bit by the appropriate power of 2

and add them together.

? ? 128 64 32 16 8 4 2 1

1 0 0 0 0 0 1 1

1 0 1 0 0 1 1 0 0

100000112 = ……………….10 ?

1010011002 = ……………………10 ?


Binary Inputs and Outputs◊ Number Representation - Binary to decimal◊ A decimal number can be converted to binary by repeated division by 2

number /2 remainder

155 77 1 Least Significant Bit

77 38 1

38 19 0

19 9 1

9 4 1

4 2 0

2 1 0

1 0 1 Most Significant bit

15510 = 100110112


Decimal to Binary

An alternative way is to use the “placement” method

128 goes into 155 once leaving 27 to be placed

So 64 and 32 are too big (make them zero)16 goes in once leaving 11

and so on…

128 64 32 16 8 4 2 1

1

1 0 0 1


Representations

◊ There are different ways of representing decimal numbers in a binary coding

◊ BCD or Binary Coded Decimal is one example.

◊ Each decimal digit is replaced by 4 binary digits



◊ 6 of the possible 16 values unused

◊ example 45310 = 0100 0101 0011BCD

◊ Note that BCD code is longer than a direct representation in natural binary code:

◊ 453 = 111000101

Decimal BCD0 00001 00012 00103 00114 01005 01016 01107 01118 10009 1001



◊ Hexadecimal and Octal◊ Writing binary numbers as strings of 1s and 0s can

be very tedious◊ Octal (base 8) and Hexadecimal (base 16) notations

can be used to reduce a long string of binary digits.

octal 512 (83) 64 (82) 8 (81) 1 (80)

1 2 0 7 512 + 128 + 7

hexadecimal 256 (162) 16 (161) 1 (160)

1 A F 256 + 160 + 15Notice that hexadecimal requires 15 symbols (each number system needs 0 – base-1 symbols) and therefore A – F are used after 9.


Octal as shorthand for Binary

◊ Each octal digit corresponds to 3 binary bits

binary octal

000 0

001 1

010 2

011 3

100 4

101 5

110 6

111 7

To convert a binary string: 10011101010011

Split into groups of 3:

010 011 101 010 011

2 3 5 2 3

Thus 100111010100112 = 235238


Similarly with Hexadecimal

◊ Each hex digit corresponds to 4 binary bits

binary hex

0000 0

0001 1

0010 2

0011 3

0100 4

0101 5

0110 6

0111 7

To convert a binary string: 10011101010011

Split into groups of 4:

0010 0111 0101 0011

Thus 100111010100112 = ……………

16 ?

binary hex

1000 8

1001 9

1010 A

1011 B

1100 C

1101 D

1110 E

1111 F


Binary inputs and outputs

◊ Colour codes◊ You often see hex used in graphic

design programs for the red, blue and green components of a colour:

◊ FF0000 represents red, for example.◊ How many bits are used to represent

each colour?◊ How many different colours can be

represented?


Binary Inputs and Outputs◊ Characters

◊ Three main coding schemes used: ASCII (widespread use), EBCDIC (not used often) and UNICODE (new)

◊ ASCII table (in hex) : 00nul

01soh

02sot

03 etx

04 eot

05enq

06 ack

07 bel

08 bs

09 ht

0a nl

0b vt

0c np

0d cr

0e so

0f si

10 dle

11 dc1

12 dc2

13 dc3

14 dc4

15 nak

16 syn

17 etb

18 can

19 em

1a sub

1b esc

1c fs

1d gs

1e rs

1f us

20 sp

21 !

22 "

23 #

24 $

25 %

26 &

27 '

28 (

29 )

2a *

2b +

2c ,

2d -

2e .

2f /

30 0

31 1

32 2

33 3

34 4

35 5

36 6

37 7

38 8

39 9

3a :

3b ;

3c <

3d =

3e >

3f ?

40 @

41 A

42 B

43 C

44 D

45 E

46 F

47 G

48 H

49 I

4a J

4b K

4c L

4d M

4e N

4f O

50 P

51 Q

52 R

53 S

54 T

55 U

56 V

57 W

58 X

59 Y

5a Z

5b [

5c \

5d ]

5e ̂

5f _

60 ̀

61 a

62 b

63 c

64 d

65 e

66 f

67 g

68 h

69 i

6a j

6b k

6c l

6d m

6e n

6f o

70 p

71 q

72 r

73 s

74 t

75 u

76 v

77 w

78 x

79 y

7a z

7b {

7c 7d }

7e ~

7f del


Gray Codes ◊ Other codes exist for specific

purposes◊ Gray codes provide a sequence

where only one bit changes for each increment

◊ Allows increments without ambiguity due to bits changing at different times.◊ E.g. changing from 3 to 4, normal

binary has all three bits changing 011 -> 100. Depending on the order in which the bits change any intermediate value may be created.

Dec Gray

0 000

1 001

2 011

3 010

4 110

5 111

6 101

7 100


Summary

◊ Support website◊ Analogue and Digital◊ Binary Number Systems◊ Coding schemes considered were:

◊ Natural Binary ◊ BCD◊ Octal representation◊ Hexadecimal representation◊ ASCII


Exercises

◊ You should practice conversions between binary, octal, decimal and hexadecimal.

◊ You should be able to code decimal to BCD (and BCD to decimal).

◊ You should be able to explain and give examples of digital and analogue data.


Binary, Hex, & BCD:Binary Number System◊ Most digital systems deal with groups of

bits in even powers of 2, such as 8, 16, 32, and 64 bits

◊ 8-bit Binary number - weighted values of each bit

27 26 25 24 23 22 21 20

128 64 32 16 8 4 2 1


Binary, Hex, & BCD:Binary Number System

◊ Example: Convert 1011 1010 to its decimal equivalent

27 26 25 24 23 22 21 20

128 64 32 16 8 4 2 1

1 0 1 1 1 0 1 0

128 + 0 + 32 + 16 + 8 + 0 + 2 + 0 = 18610


Binary, Hex, & BCD:Binary Number System

◊ Fractional Binary Numbers◊ Example: Convert 1011.1010 to its

decimal equivalent

23 22 21 20 2-1 2-2 2-3 2-4

8 4 2 1 1/21/4

1/81/16

1 0 1 1 1 0 1 0

8 + 0 + 2 + 1 + 0.5 + 0 + 0.125 + 0

= 11.62510


Review – Decimal to Binary

Example 3810 = _______21001102 38

2 9Quotient

Quotient

0

1

1

Remainder

Remainder

RemainderLSB

2 19

2 4

Quotient

0

2 2

Quotient

0

2 1 1 MSB

Remainder

Remainder

Remainder


Review – Decimal to Binary

Example 3810 = _______2100110

27 26 25 24 23 22 21 20

128 64 32 16 8 4 2 1

0 0 1 0 0 1 1 0

0 + 0 + 32 + 0 + 0 + 4 + 2 + 0

= 3810


Binary, Hex, & BCD:Hexadecimal Number System

◊ Most digital systems deal with groups of bits in even powers of 2, such as 8, 16, 32, and 64 bits

◊ Hexadecimal uses groups of 4 bits◊ Base 16

◊ 16 possible symbols◊ 0 thru 9 and A thru F

◊ Easier handling of long binary strings


Binary, Hex, & BCD:Converting Hex to Decimal◊ Multiply each digit by its positional

weight

Example: 24316 = 2 x (162) + 4 x (161) + 3 x

(160)= 512 + 64 + 3= 57910


Binary, Hex, & BCD:Converting Decimal to Hex

◊ Use repeated division method◊ Divide decimal number by 16◊ First remainder is LSB; last is MSB

◊ Note: when done on calculator, the fractional portion can be multiplied by 16 to get the remainder


Example 57910 = _____16243

16 579

16 36

16 2

0

Quotient

Quotient

Quotient

3

4

2

Remainder

Remainder

RemainderLSD

MSD


Binary, Hex, & BCD:Converting Hex to Binary

◊ Example:9F216 = 9 F 2

= 1001 1111 0010 = 1001111100102


Binary, Hex, & BCD:Converting Binary to Hex

◊ Group bits in fours starting with LSB◊ Convert each group to hex digit◊ Add leading zeros to left of MSB of

last group, as needed


Binary, Hex, & BCD:Converting Binary to Hex

◊ Example:11101001102 =0011 1010 0110

= 3 A 6 = 3A616

◊ Counting in hex reset & carry after F


Number Systems Conversion Table

Decimal Binary Octal Hexadecimal BCD

0 0 0 0 0

1 1 1 1 0001

2 10 2 2 0010

3 11 3 3 0011

4 100 4 4 0100

5 101 5 5 0101

6 110 6 6 0110

7 111 7 7 0111

8 1000 10 8 1000

9 1001 11 9 1001

10 1010 12 A 0001 0000

11 1011 13 B 0001 0001

12 1100 14 C 0001 0010

13 1101 15 D 0001 0011

14 1110 16 E 0001 0100

15 1111 17 F 0001 0101


TTL Logic Chips-Basic Circuitry

◊ Power & Ground (on basic gates - 14 pin DIP)◊ Pin 14 – Vcc (+5V)◊ Pin 7 – GND

◊ Absolute limits7400 NAND Gate Data Sheet

74LS00


TTL 74 Series Logic Chips◊ Logic Level voltage ranges

◊ VIN High = 2.0Vmin◊ VIN Low = 0.8Vmax◊ VOUT High = 2.4Vmin◊ VOUT Low = 0.4V max

◊ Indeterminate voltagesAny voltage between 0.8V and 2.0V on an

input can not be guaranteed to be either high or low

◊ Current capabilities◊ IIN High = 40uA◊ IIN Low = -1.6mA◊ IOUT High = -0.4mA◊ IOUT Low = 16mA


TTL 74 Series Logic Chips

◊ Fan outFan out (HIGH) = IOH(max) / IIH(max)

For 7400: 400uA/40uA = 10Fan out (LOW) = IOL(max) / IIL(max)

For 7400: 16mA/1.6mA = 10Propagations delays

◊ How long does it take the output to change after a change has happened at the inputs


Data Sheets

◊ Floating inputs◊ What happens if you don’t connect an

input to a high or low

◊ CMOS family◊ MOSFET switches instead of bipolar

junction transistor switches◊ Faster than most standard TTL chips◊ More susceptible to static electricity


A

BY

Review of Logic Functions

1AND2

A B Y

0 0 1

0 1 1

1 0 1

1 1 0

NAND GateTruth Table

NAND: AB A × B A & B



1AND2

A B Y

0 0 1

0 1 0

1 0 0

1 1 1

XNOR GateTruth Table

XNOR: A + B A $ B

A

BY



1AND2

A B Y

0 0 0

0 1 1

1 0 1

1 1 1

OR GateTruth Table

OR: A + BA # B

A

BY



1AND2

A B Y

0 0 0

0 1 0

1 0 0

1 1 1

AND GateTruth Table

A

BY

AND: AB A × B A & B



1AND2

A B Y

0 0 1

0 1 0

1 0 0

1 1 0

NOR GateTruth Table

OR: A + BA # B

A

BY



1AND2

A B Y

0 0 0

0 1 1

1 0 1

1 1 0

XOR GateTruth Table

XOR: A + B A $ B

A

BY



A Y

0 1

1 0

NOT

A

!

NOT Gate

Truth Table

A Y


52

Review

◊ There are 3 basic digital gates:

◊ AND

◊ OR

◊ NOT


53

Review

AND, where ALL inputs must be “1” for the output to be “1”

OR, where ANY of the inputs can be “1” for the output to be “1”

NOT (or the Inverter) where the output is the opposite (compliment) of the input.


54

Review Questions

What is the outcome of the following:

1

1

0

1

0

1

1

55

Basic Digital Logic 2

Basic Combinational Logic, NAND and NOR gates


56

Combinational Logic

◊ A circuit that utilizes more that 1 logic function has Combinational Logic.

◊ As an example, if a circuit has an AND gate connected to an Inverter gate, this circuit has combinational logic.


57

Combinational logic

◊ How would your describe the output of this combinational logic circuit?


58

NAND Gate

◊ The NAND gate is the combination of an NOT gate with an AND gate.

The Bubble in front of the gate is an inverter.


59

Combinational logic

◊ How would your describe the output of this combinational logic circuit?


60

NOR gate

◊ The NOR gate is the combination of the NOT gate with the OR gate.

The Bubble in front of the gate is an inverter.


61

NAND and NOR gates

◊ The NAND and NOR gates are very popular as they can be connected in more ways that the simple AND and OR gates.


62

Exercise 1 – a bit challenging

Complete the Truth Table for the NAND and NOR Gates

Input Output

0 0

0 1

1 0

1 1

Input Output

0 0

0 1

1 0

1 1

NAND NOR

Hint: Think of the AND and OR truth tables. The outputs for the NAND and NOR are inverted.


63

Exercise 2 – more challenging

◊ Turn the NAND and NOR gates into inverter (NOT) gates. Include a switch for the input.

64

Basic Digital Logic 2

Chips and Gates


65

Basic Digital Chips

◊ Digital Electronics devices are usually in a chip format.

◊ The chip is identified with a part number or a model number.

◊ A standard series starts with numbers 74, 4, or 14.◊ 7404 is an inverter◊ 7408 is an AND◊ 7432 is an OR◊ 4011B is a NAND


66

Chips

◊ Basic logic chips often come in 14-pin packages.

◊ Package sizes and styles vary.

◊ Pin 1 is indicated with a dot or half-circle

◊ Numbers are read counter-clockwise from pin 1 (viewed from the top)

Pin 1 Pin 7

Pin 14 Pin 8


67

Chips

◊ Chips require a voltage to function

◊ Vcc is equal to 5 volts and is typically pin 14

◊ Ground is typically pin 7

Pin 1 Pin 7

Pin 14 Pin 8


68

Chips – Specification Sheet

Diagram from http://www.onsemi.com

Voltage

Ground

The voltage and ground pins must be connected for the device to function. Check the specification sheet to make sure.


69

Chips – Specification Sheet

Diagrams from http://www.onsemi.com

A

B

C

D


70

Wiring a chip

Vcc

Vcc

IN

IN

OUT

Probe


71

Lab Exercise

◊ Using the experimenter’s boards, connect the circuit provided to you in the following pages.


72

Layout of the SK-10 Experimenter's Board


73

Layout of the SK-10 Experimenter's Board

Flat Side

Wires

7400


74

Logic Diagram of Lab


75

Useful Resources◊ Textbooks on Digital Electronics (used is ok!)◊ Electronics Workbench or other electronic simulation

software◊ Craig Maynard’s Virtual Vulcan◊ The following web sites:

◊ http://learnat.sait.ab.ca/ict/digi240_godin/default.htm◊ http://learnat.sait.ab.ca/ict/cmph200/Default.htm◊ http://learnat.sait.ab.ca/ict/cmph200_godin/default.htm

◊ http://focus.ti.com/docs/logic/logichomepage.jhtml◊ http://www.onsemi.com◊ http://www.national.com/

◊ http://www.play-hookey.com/digital/◊ http://www.crhc.uiuc.edu/~drburke/databookshelf.html◊ http://www.digikey.ca/

Documents

1 Basic Digital Logic. Prof Jess Role @UEAB 2008 ◊Digital Electronics ◊Number Systems and Logic ◊Electronic Gates ◊Combinational Logic ◊Sequential Circuits