Digital Logic Circuit Design Putting logic to use

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Digital Logic Circuit DesignPutting logic to use
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Introduction

• So you know and love the fundamental logic gates
• But why are they good for?
• How are they used in real life?

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Digital Design

• When we teach combinational circuits, often, the
  circuit comes before the truth table

• But this is backwards to reality
• In circuit design, we develop a truth table and
  then use it to determine the circuit needed

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The Design Process

• Suppose you want to design an electronic circuit
  for a 2-person voting system that determines if
  there is a consensus
• How do you go about this?
• What steps are involved?

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Step 1 Declaring Variables(the hard part)

• When working with digital logic we must use
  Boolean values (on/off or 0/1)
• The first step is to model the systems inputs
  and outputs as Boolean values
• Each input will be a separate variable
• The output will be a separate variable
• The variables must be chosen so they can be
  represented as Boolean values

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Step 1 Declaring Variables(the hard part)

• Inputs will be
• The first voter, call it A
• The second voter, call it B
• Either of which can be either yes/no values
• The output can be called Y, with yes meaning a
  consensus and no meaning not a consensus

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Step 2 Determine the Truth Table

• The next step is to determine the truth table -
  that is, what combinations of inputs make our
  output(s) true (i.e. 1)
• In our case the truth table is as follows

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Step 2 Determine the Truth Table
Two different versions of the truth table
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Step 3 Write the Expression

• Logic functions derived from a truth table can be
  very complex
• The Boolean logic functions derived are called
  minterm expressions
• These functions are the sum of products of
  Boolean variables that have an output value of
  true

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Step 3 Write the Expression

• Only rows with the output 1 have minterms written
• The minterms are summed together to give an
  expression for Y

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Step 3 Write the Expression

• In this case our Boolean expression would be
• The 2 values added (that is ored) together
  correspond to the expressions for the rows in the
  truth table with 1s
• These expressions are called minterms
• http//doyle.wcdsb.ca/ICE4MI/digitial_electronics/
  minterms.htm

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Step 4 Simplification

• Minterm expressions can be simplified using
  Boolean Algebra Laws or Karnaugh Maps (Kmaps)
• For example, the expressionsimplifies to
• This is because it is true only in all of the
  cases when B is true
• Advantages to simplification include economics,
  clarity and aesthetics

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Step 4 Simplification with Boolean Algebra Laws

• And Laws
• A1A
• A00
• AAA
• AA0
• ABBA
• (AB)CA(BC)
• A(BC)(AB)(AC)
• (AB)AB

• Or Laws
• A11
• A00
• AAA
• AA1
• ABBA
• (AB)CA(BC)
• A(BC)(AB)(AC)
• (AB)AB

• http//doyle.wcdsb.ca/ICE4MI/digitial_electronics/
  boolean_algebra_laws.htm

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Step 4 Simplification with Boolean Algebra

• Using the rules on the last slide show that
• AB AB B
• Left Side
• AB AB
• (A A) B
• 1 B
• B

• http//doyle.wcdsb.ca/ICE4MI/digitial_electronics/
  boolean_simplification.htm

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Step 4 Simplification with K-map

• A K-map is an alternate format for a truth table
• Simplification becomes mechanical, easy
• Step 1 draw a k-map
• Variables go on sides
• Place outputs of 1 at intersections

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Step 4 Simplification with K-map

• Step 2 draw loops
• All 1s must be in at least one loop
• Loops can contain 1, 2, 4, 8 1s
• Loops may be created by going off the side, top
  or bottom of the k-map (the k-map wraps around)
• Any variable that appears with its complement in
  a loop is eliminated
• Write a minterm for each loop

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Step 4 Simplification with K-map

• Step 2 draw loops
• One loop contains 2 1s
• The variable A and its complement appear in the
  loop, so they are eliminated
• The expression is

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Step 4 Simplification continued

• Complex Boolean expression simplification can
  also be done using software
• http//doyle.wcdsb.ca/ICE4MI/digitial_electronics/
  karnaugh_maps.htm
• Simple Kmap programs exist as well
• http//doyle.wcdsb.ca/ICE4MI/digitial_electronics/
  KarnaughExplorer.htm

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Step 5 Build or Prototype the circuit

• At this point you are ready to simulate the
  circuit using software or create your circuit
  using logic chips and input/output components
• For the voting system
• Simplification does not yield anything simpler
• inputs can be simple solid state, on/off switches
• the logic is a combination of AND and OR gates
• outputs can be shown with LEDs

• http//doyle.wcdsb.ca/ICE4MI/LearnAndOrNot/index.h
  tml

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Step 5 Creating the circuit

• You could have students build their circuits into
  a working model
• Ideas include traffic light systems, voting
  systems, games, alarm/sensor systems
• Individualized assignments (with the same answers
  for easy evaluation)
• Turn on motor to close the garage door when the
  sun sets or its raining and the door is up
• Give a parking ticket when the meter has run out
  or car not parked correctly and the car is red

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Your Turn

• In a group of 2-4 people, design the logic
  circuit for one of the following
• A 2 person voting system with 3 outputs majority
  for, against and tie
• A 3 person voting system with 2 outputs for and
  against
• A walk signal for a standard traffic light
• A circuit that compares two 2-bit values and
  outputs if they are the same

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Design Resources

• Reid, Neal E. and Wilson, Stanley L. Computer
  Science Program Design and Technology. Toronto
  John Wiley Sons, 1985, pp 334-365.

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Minterm Resources

• Minterms http//doyle.wcdsb.ca/ICE4MI/digitial_el
  ectronics/minterms.htm
• Simplification with Karnaugh maps ( minterms)
  http//doyle.wcdsb.ca/ICE4MI/digitial_electronics/
  karnaugh_maps.htm
• Karnaugh map explorer http//doyle.wcdsb.ca/ICE4M
  I/digitial_electronics/KarnaughExplorer.htm

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Boolean Algebra Resources

• Boolean algebra laws http//doyle.wcdsb.ca/ICE4MI
  /digitial_electronics/boolean_algebra_laws.htm
• Boolean algebra simplification
  http//doyle.wcdsb.ca/ICE4MI/digitial_electronics/
  boolean_simplification.htm
• Logic gate simulator http//doyle.wcdsb.ca/ICE4MI
  /LearnAndOrNot/index.html