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### Basic Digital Logic 2

### Basic Digital Logic 2

- 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)

1

0

1

0

1

For example, Binary Alphabet: 0, 1.

Slide example

- Consider a child’s slide in a playground:

a set of discrete steps

continuous movement

levels

Input

Range

for 1

Output

Range

for 1

2.8

2.4

0.8

Input

Range

for 0

Output

Range

for 0

0.4

0 Volt

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

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

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

Binary Inputs and Outputs

- Both Decimal and Binary numbers use a positional weighting system, eg: 10102 = 1x23+0x22+1x21+0x20 = 1x8 + 0x4 + 1x2 + 0x1 = 1010

Binary to decimal

- Multiply each 1 bit by the appropriate power of 2 and add them together.

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

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…

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

Binary Inputs and Outputs

- 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

Binary Inputs and Outputs

- 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.

Notice 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

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

To convert a binary string: 10011101010011

Split into groups of 4:

0010 0111 0101 0011

Thus 100111010100112 = ……………16 ?

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) :

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.

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

Binary, Hex, & BCD:Binary Number System

- Example: Convert 1011 1010 to its decimal equivalent

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

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

= 11.62510

Remainder

Remainder

Remainder

Remainder

Remainder

2

2

Quotient

Quotient

Quotient

Quotient

2

2

2

2

Review – Decimal to Binary100110

Example 3810 = _______2

LSB

38

0

1

19

9

1

4

0

2

0

1

1

MSB

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

Remainder

Remainder

16

16

Quotient

Quotient

Quotient

16

Example 57910 = _____16243

579

3

LSD

36

4

2

2

MSD

0

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

TTL Logic Chips-Basic Circuitry

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

- Absolute limits
7400 NAND Gate Data Sheet

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 voltages
Any 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 out
Fan out (HIGH) = IOH(max) / IIH(max)

For 7400: 400uA/40uA = 10

Fan out (LOW) = IOL(max) / IIL(max)

For 7400: 16mA/1.6mA = 10

Propagations 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

Review

- There are 3 basic digital gates:
- AND
- OR
- NOT

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.

Basic Combinational Logic, NAND and NOR gates

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.

Combinational logic

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

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.

Combinational logic

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

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.

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.

Exercise 1 – a bit challenging

Complete the Truth Table for the NAND and NOR Gates

NOR

NAND

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

Exercise 2 – more challenging

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

Chips and Gates

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

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 14

Pin 8

Pin 7

Pin 1

Chips

- Chips require a voltage to function
- Vcc is equal to 5 volts and is typically pin 14
- Ground is typically pin 7

Pin 14

Pin 8

Pin 7

Pin 1

Chips – Specification Sheet

Voltage

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

Ground

Diagram from http://www.onsemi.com

Lab Exercise

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

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/

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