ECED2200 Digital Circuits Introduction, Gates, Number Systems - - PowerPoint PPT Presentation

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ECED2200 Digital Circuits Introduction, Gates, Number Systems 05/07/2012 Colin OFlynn - CC BY-SA 1 General Notes See updates to these slides: www.newae.com/teaching These slides licensed under Creative Commons

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SLIDE 1

ECED2200 – Digital Circuits

Introduction, Gates, Number Systems

05/07/2012 Colin O’Flynn - CC BY-SA 1

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SLIDE 2

General Notes

  • See updates to these slides: www.newae.com/teaching
  • These slides licensed under ‘Creative Commons Attribution-ShareAlike 3.0

Unported License’

  • These slides are not the complete course – they are extended in-class
  • You will find the following references useful, see

www.newae.com/teaching for more information/links:

– The book “Bebop to the Boolean Boogie” which is available to Dalhousie Students – Course notes (covers almost everything we will discuss in class) – Various websites such as e.g.: www.play-hookey.com – The book “Contemporary Logic Design”, which was used in previous iterations of the class and you may have already

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SLIDE 3

INTRODUCTION, BINARY, AND GATES

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Analog vs. Digital

Analog (Infinite Values) Digital (Discrete Values) time time

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SLIDE 5

Digital Systems

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Binary

Off False Low On True 1 High

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Binary – Waveforms in Time

1

1 1 1

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GATES

Input #1 Input #2 Output

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OR Gate – Truth Table

Input #1 Input #2 Output 1 1 1 1 1 1 1

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OR Gate

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OR Gate

1 1

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OR Gate

1 1

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OR Gate

1 1 1

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OR Gate

A B Y 1 1 1 1 1 1 1

Y=A+B

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AND Gate

A B Y 1 1 1 1 1

Y=A B

  • 05/07/2012

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NOT Gate

A Y 1 1

Y=A

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Example!

IN1 IN2 OUT 1 1 1 1 1 IN1 OUT 1 1 IN1 IN2 OUT 1 1 1 1 1 1 1 AND NOT OR A B Y 1 1 1 1

Fill in this table:

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Example!

What is the Boolean function of the above schematic?

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Example!

Y=A•B+A•B

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Little Circles

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Not AND = NAND Gate

A B Y 1 1 1 1 1 1 1

Y=A B

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Not OR = NOR Gate

A B Y 1 1 1 1 1

Y=A+B

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Supergates

All basic logic operations can be formed with NAND gates (or NOR gates).

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Example!

A B Y 1 1 1 1 1 1 1 A B Y 1 1 1 1 1 1 1

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Example!

A B Y 1 1 1 1 1 1 1

A B Y 1 1 1 1 1 1 1

If we inverted each of the A & B inputs to the NAND gate, note we get the same truth table as the OR gate!

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Example!

A B Y 1 1 1 1 1 1 1

Y=A•B=A+B

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Example!

A B Y 1 1 1 1 1 1 1

Y=A•B=A+B

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Why Do you Care?

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Example – AND Gate

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Exclusive OR (X-OR) Gate

A B Y 1 1 1 1 1 1

Y=A B ⊕

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XOR Gate Implementation

A B=(A+B)•(A•B)=A•B+A•B ⊕

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Exclusive NOR (X-NOR) Gate

A B Y 1 1 1 1 1 1

Y=A B 

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Section Summary – 1/3

Gates have inputs & outputs. Inputs are binary (Boolean) variables with two possible values ‘1’ (True) or ‘0’ (False). The ‘Truth Table’ is a table showing every possible input & the resulting output. Different gates have different truth tables.

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Section Summary – 2/3

The following are the basic gates:

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Section Summary – 3/3

Based on the truth tables, one can make any of the gates with NAND (or NOR) gates. E.g. making an OR gate with NAND gates:

A B Y 1 1 1 1 1 1 1

Y=A•B=A+B

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References / Reading

  • ECED2200 Notes, “Digital Circuits” section
  • Bebop to the Boolean Boogie, Chapter 5
  • CLD, Chapter 1

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PHYSICAL GATE IMPLEMENTATION

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Switch Logic

OR Gate AND Gate NOT Gate

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Switch NOT

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Switch NOT

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Switch NOT

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Field Effect Transistor Switch

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FET Logic Gates – Inverter (NOT)

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FET Logic Gates – Inverter (NOT)

PMOS NMOS

CMOS = Complementary MOS (e.g.: uses both positive & negative MOS)

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FET Logic Gates - NAND

Source: http://commons.wikimedia.org/wiki/File:NAND_gate_(CMOS_circuit).PNG

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FET Logic Gates - NOR

http://commons.wikimedia.org/wiki/File:NOR_gate_%28CMOS_circuit%29.PNG

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Logic Families (Types)

  • 1. Diode Logic (DL)
  • 2. Resistor-Transistor Logic (RTL)
  • 3. Diode-Transistor Logic (DTL)
  • 4. Transistor-Transistor Logic (TTL)
  • 5. Metal-Oxide Semiconductor (MOS)
  • 6. Complementary MOS (CMOS)
  • 7. Emitter-Coupled Logic (ECL)
  • 8. BiCMOS

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Characteristics of Logic Types

  • Fan-in
  • Fan-out
  • Speed
  • Noise Margin
  • Power
  • Size

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Section Summary

  • Bebop to the Boolean Boogie Chapter 6
  • ECED2200 Notes “Electric Switches + Logic

Classifications”

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NUMBER SYSTEMS

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Binary to Decimal

1110 1011 1 1 1 1 1 1 128 + 64 + 32 + + 8 + + 2 + 1 = 235 Decimal

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Let’s Do a Hand-out: INTRO-1 (Top)

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Decimal to Binary

216 Decimal

216 – 128 = 88 (1 in 27 position)

1 1

88 – 64 = 24 (1 in 26 position) 24 – 32 < 0, so 0 in 25 position

1

24 – 16 = 8 (1 in 24 position) 8 – 8 = 0 (1 in 23 position) 0 in remaining positions

1

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Let’s Do a Hand-out: INTRO-1 (Bottom)

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Number Notation

There are 10 kinds of people in the word – those that understand binary, and those that don’t.

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Number Notation

Ambiguity unacceptable - we are engineers not comics. 102 = 10B = 210 (2 decimal) 1010 = 10D = 10102 (1010 binary) e.g.: Conversion: 47210 = 111011000

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How to Check your Conversions

Windows Calculator (Windows 7 version shown)

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How to Check your Conversions

Stand Alone Calculator

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Other Number Systems: Hex & Octal

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Cheat Sheet

Decimal Binary Hexadecimal (0x) Octal (0) 0000 1 0001 1 1 2 0010 2 2 3 0011 3 3 4 0100 4 4 5 0101 5 5 6 0110 6 6 7 0111 7 7 8 1000 8 10 9 1001 9 11 10 1010 A 12 11 1011 B 13 12 1100 C 14 13 1101 D 15 14 1110 E 16 15 1111 F 17 05/07/2012 Colin O’Flynn - CC BY-SA 60

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Advantages of Hex (and Octal)

1010 1000 1101 1111 1011 00102 This is only 24 bits – but long/hard to write… Equivalent to 1106731410 Easier to write, but conversion is error-prone, plus we are normally lazy…

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Advantages of Hex

1010 1000 1101 1111 1011 00102 A 8 D F B 2 = 0xA8DFB2 Much easier to write, easy to convert!

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Advantages of Octal

101 010 001 101 111 110 110 0102 5 2 1 5 7 6 6 2 = 052157662 in Octal Easy to write (longer than hex though), still easy to convert!

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Binary Coded Decimal

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Section Summary

  • Bebop to the Boolean Boogie: Chapter 7
  • ECED2200 Notes: “Number Systems”

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ADDING & SUBTRACTING IN BINARY

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Adding in Binary

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Subtracting in Binary

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Positive & Negative Numbers

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Number Wheel

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0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

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Disadvantage of Sign Magnitude

  • Two Zeros
  • Positive & Negative numbers require different

processing or else addition is wrong: (-5) + (+1) = 1101 + 0001 = 1110 = -6 (-3) + (-2) = 1011 + 1010 = 0101 = +5

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Two Zeros? Where did Zero even come from?

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Rules of Brahmagupta (628 AD):

– The sum of zero and a negative number is negative. – The sum of zero and a positive number is positive. – The sum of zero and zero is zero. – The sum of a positive and a negative is their difference; or, if their absolute values are equal, zero.

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Complementary Numbers

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Diminished Radix Complement

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Radix Complement

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2’s Complement

1 1 0 1 0 0 0 1 0 0

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Instructions

  • 1. Form 1’s complement by inverting all bits
  • 2. Add 1 to 1’s complement to get 2’s

complement

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2’s Complement

1 1 0 1 0 0 0 1 0 0

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Bit Notations

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Instructions:

  • 1. Starting from Least Significant Bit (LSB)

working towards Most Significant Bit (MSB), copy number one bit at a time.

  • 2. Continue copying until you reach the first ‘1’.

Copy this ‘1’ bit as-is

  • 3. For remaining bits invert them (0->1, 1->0)

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Some Examples

1 0 1 0 0 1 0 1 0 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1

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Number Wheel

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0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

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Section Summary

  • Bebop to the Boolean Boogie: Chapter 8
  • Contemporary Logic Design:
  • ECED Notes: “Number Systems”

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