LECTURE 13 DIGITAL ELECTRONICS

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LECTURE 13 DIGITAL ELECTRONICS
Dr Richard ReillyDept. of Electronic
Electrical EngineeringRoom 153, Engineering
Building
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Analysis of Combinational Logic

• Examples of combinational circuits
• decoders, encoders, multiplexers, adders,
  subtractors, multipliers, comparators, etc.
• Need to consider the implementation of
  combinational systems with combinational logic
  circuits.
• Combinational logic deals with the method of
  combining basic gates into circuits that carry
  out a desired application.

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Combinational Circuits

• DEFINITION
• Logic circuits without feedback from output to
  input, constructed from a functionality complete
  gate set are said to be combinational .
• Logic circuits that contain no memory (ability to
  store information) are combinational.
• Those that contain memory, including flip-flops
  are said to be sequential

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Mathematical definition

• Let X be the set of all input variables
• and Y set of all output variables.
• ? The combinational function F operates on input
  variables set X to produce output variable set Y
• Output variables are not fed back to the input.

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General Logic Design Sequence

• Problem Statement
• Truth-Table Construction
• Switching Equations Written
• Equations Simplified
• Logic Diagram Drawn
• Decide on Logic Family for Implementation
• Logic Circuit Built

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Derivation of Switching Equation

• Logic can be described in several ways
• Truth Table
• Logic Diagram
• Boolean Equation

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Combinational Logic

• Each input variable group that produces a logical
  1 in a truth table output column can form a term
  in an Boolean Expression.
• Each term is formed by ANDing input variables
• Each AND term is then ORed with other AND terms
  to complete output Boolean Equation
• NOTE
• Each AND term (also called a product term)
  identified one input condition where the output
  is a logical 1.

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Definitions

• Literal
• A Boolean variable or its complement.
• e.g. ? and are both literals
• Product Term
• A product term is a literal or the logical
  product (AND) of multiple literals.
• e.g. Let be binary variables
• ? a product term could be

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• Sum Term
• A sum term is a literal or the logical OR of
  multiple literals.
• e.g. Let be binary variables ? a sum
  term could be
• Sum of Products
• SOP is the logical OR of multiple product terms.
  Each product term is the AND of binary literals.
• e.g. is a SOP expression
• Products of Sums
• POS is the logical AND of multiple OR terms. Each
  sum term is the OR of binary literals.
• e.g.
  is a POS expression

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Minterms and Maxterms

• Minterm
• A minterm is a special case product (AND) term.
• A minterm is a product term that contains all the
  input variables (each literal no more than once)
  that make up a Boolean expression.
•  
• Maxterm
• A maxterm is a special case (OR) term.
• A maxterm is a sum term that contains all the
  input variables (each literal no more than once)
  that make up a Boolean expression.

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Canonical Sum of Products

• A canonical SOP is a complete set of minterms
  that defines when an output variable is a logical
  1.
• Each minterm corresponds to the row in the truth
  table when the output function is 1.

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Canonical Product of Sums

• A canonical POS is a complete set of maxterms
  that defines when an output variable is a logical
  0.
• Each maxterm corresponds to the row in the truth
  table when the output function is 0.

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Canonical Forms

• Canonical defined as conforming to a general
  rule.
• The rule for switching logic in that each term
  used in a switching equation must contain all of
  the variables.
• Two formats generally exist for expressing
  switching equations in a canonical form.
• Sum of minterms
• Product of maxterms

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Canonical Forms

• Canonical forms are not simplified
• Normally the opposite of simplification,
  containing redundancies.
• Use Boolean Theorems to simplify the expressions
• to eliminate redundancy
• lower cost of the final logic circuit
• Design may require converting to logic realised
  in one form to another form
• TTL NAND gates to ECL NOR gate
• Thus can be better to convert to canonical form
  before simplification carried out

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Generation of Switching Equations from Truth-Table

• What happens when we have a large number of
  minterms or maxterms ?
• Switching equations can be written more
  conveniently by using minterm or maxterm
  numerical designation.
• where decimal equivalent value for the term can
  be written directly.

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Generation of Switching Equations

• If decoded each of the minterms based on binary
  weighting of each variable and produce a list of
  decimal minterms, the result would be

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Generation of Switching Equations

• A canonical POS is representation by

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Simplification of Boolean Expressions

• Prime Implicants
• Essential Prime Implicants

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Irish Times, Friday 9th January
Boolean Expression for such systems tend to be
composed for many variable.
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Simplification of Boolean Expressions

• For many applications the number of variables in
  a problem is too large to simplify normally using
  Karnaugh Maps.
•  
• When the number of variables approaches 6 or 7,
  the map system becomes unworkable and prone to
  errors.
• Simplification means that the logic designer can
  obtain more fundamental use from a given
  component.
• ? a software method would be more efficient when
  the number of variable is large.

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The Quine-McCluskey method

• Algorithm that uses the same Boolean Algebra
  postulates (as with Karnaugh Map) but in a form
  suitable for a computer solution.
•  
• Large Karnaugh Maps require recognition of groups
  of terms that may form essential prime
  implicants.
• Problem
• The larger the map the more difficult this
  pattern recognition task becomes !!

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The Quine-McCluskey Example

• Minimise D using the Quine-McCluskey method
•  
• Arrange all minterms in a list of increasing
  order, so that groups of terms contain the same
  number of 1s

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• Write out Binary Equivalent
• 0000
• 0001
• 0010
• 0011
• 0110
• 0111
• 1000
• 1001
• 1110
• 1111

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• Arrange all minterms in a list of increasing
  order, so that groups of terms contain the same
  number of 1s
• Table-1

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• Table-1

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The Quine-McCluskey method

• Group 0 contains no 1s
• Group 1 contains only minterms that have a single
  1 1,2,8
• Group 2 contains only minterms that have a two
  1s 3,6,9
• Group 3 contains only minterms that have a three
  1s 7,14
• Group 4 contains only minterms that have a four
  1s 15
•  

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The Quine-McCluskey method

•  
• The relation can be applied to
  pairs of minterms where only one variable changes
  value.
• Candidates for this relationship can come from
  adjacent groups of minterms

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Step 2

• Create a new table showing the minterms in group
  that matched with those from group
  such that they differ in only one
  position.This is equivalent to or
  Eliminated variable bit positions are
  indicated by the - symbol.
• Þ

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Step 2

• What do you do ?
• Compare minterm 0 group 0, with each minterms
  in group 1.
• Minterm 0 (group 0) and Minterm 1 (group 1)
  differ by only one variable giving

• SimilarlyMinterm 0 (group 0) can combine with
  minterms 2 and 8 in group 1, producing
  (Minterms 0,2) (Minterms 0,8)
• The combination of minterms (0,1)(0,2)(0,8)
  forms group 0 in a new table, Table-2

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The Quine-McCluskey method

Table-2
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Table-2
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The Quine-McCluskey method

• As each minterm, from a group in Table-1,
  combines with a minterm in the next higher group
  it is ticked off, indicating that it is now part
  of a larger group.
• If a minterm did not combine with another, then
  no mark would be made.
• If a term does not simplify it is a prime
  implicant.

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No prime implicantsin this case

Table-1
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Step 3

• All of the adjacent minterm groups contained in
  Table-2 are compared to see if groups of four can
  be made..i.e. Table-3
• Criteria for forming groups of four
• Compare minterms in group that matched
  with those in group .
• Those that meet the criteria are combined in a
  larger group.
• Minterms 0,1 in group 0 combines with minterms
  2,3 in group 1 to form a minterm 0,1,2,3

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Step 3

• If the symbols are in the same position and
  only one other variable changes, then a new,
  larger group is created and entered into a new
  table, Table-3.
• Table-3

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Table-2
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The Quine-McCluskey method

• Those minterm groups that are used in the
  forming of larger groups are ticked off and any
  unticked groups are prime implicants.

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The Quine-McCluskey method

• Repeat step 3, with the criteria that both
  symbols must be in the same position with only
  one variable allowed to change.
• Same process repeated until no other combinations
  of minterms is possible
• 5. All unticked minterms are now considered to
  be prime implicants
• All prime implicants are formed into a
  prime-implicant table, Table-4.

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The Quine-McCluskey method

• Minterms
• Table-4

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The Quine-McCluskey method

• Table- 3

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The Quine-McCluskey method

• Minterms
• Table-4

x
x
x
x
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The Quine-McCluskey method

• Evaluate the prime implicants by circling those
  minterms that are contained in only one prime
  implicant
• i.e. only one x in a column
• NOTE
• minterms (8,9,14,15) meet this condition
• circle i.e.
• Circled minterms are Essential prime implicants

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The Quine-McCluskey method

• Minterms
• Table-4

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What are the Essential Prime Implicants ?

• Minterms 0,1,8,9 and 6,7,14,15 are essential
  prime implicants.
• Minterms 2,3 are contained in two prime
  implicants, 0,1,2,3 and 2,3,6,7.
• Need to have one or the other of these prime
  implicants but not both.Þ can choose either
• or

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Dont Cares

• The same rules apply to the Quine-McCluskey
  method with regard to Dont Cares as with the
  Karnaugh map method.

Exercise Try Answer
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Hazards

• Boolean functions used to provide
• Description of the operation of circuits
• Minimisation of Boolean expressions to design
  simpler circuits to implement the task in
  question.
•  
• Under certain circumstances, a minimal gate
  implementation of logic function may not be
  satisfactory solution of a design problem.

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Hazards

• A glitch or logic-spike is an unwanted pulse at
  the output of a combinational logic network.
• A circuit with the potential for a glitch is said
  to have a hazard.
•  
• A hazard is something intrinsic about a circuit.
• circuit with a hazard may or may not glitch
  depending on the input patterns and the
  electrical characteristics of the circuit.
•  
• Design of hazard free circuitry is critical.

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Hazards

• Hazards a problem for digital systems in two
  cases
•  
• Time sensitive logic makes a decision based on
  the output of a function without allowing the
  output to settle to a final steady-state
  value.Solution
• increase interval between the time when inputs
  first begin to change and the time when outputs
  are examined by the decision making logic.I.e.
  increase the system clock period.

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Hazards

• Hazardous circuits when connecting to a component
  with asynchronous inputs (inputs take effect as
  soon as they change rather then when sampled with
  as reference signal).Solution
• Avoid clocked parts with asynchronous inputs.
•  
• Both solutions not always possible
• ? need methods for eliminating hazards.

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Static Hazards

• Occur when possible for an output to undergo a
  momentary transition when it is expected to
  remain unchanged.
•  
• Static-1 hazard
• occurs when output momentarily goes to 0 when it
  should remain a 1.
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• Static-0 hazard
• occurs when output momentarily goes to 1 when it
  should remain a 0.

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Dynamic hazards

• occurs when output signal has potential to change
  more than once when it is expected to make a
  single transition from 0?1 or 1?0.
• Occur when there are multiple paths with
  different delays from the inputs to the outputs
• Difficult to eliminate.

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Dynamic hazards

• Example
• Consider the operation when B C 1 and A
  changes state.
•  
• is slightly delayed from A because of inverter
  gate.

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Dynamic hazards

• During a change from 1 ? 0
• extra gate causes both A and to be at a 0
  transiently.
• ? results in a transient 0 output.

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Removal of hazards

• Static hazards are eliminated as follows
• (assuming only one input changes).
• Function expanded using Canonical Expansion
• Mapped onto a Karnaugh Map
• Minterms are formed, to specify minimal
  solution.

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Removal of hazards

• It can be observed that when A goes from 1 ?
  0? a static-1 hazard occurs.

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Removal of hazards

• 4. Minterms that do not overlap any other minterm
  groups are identified as potential hazards 
•  
• 5. Additional minterms are introduced crossing
  over the existing minterms so that no minterm
  group is not overlapped by another group.

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Dynamic hazards

• Extra minterm BC is introduced between the two
  original minterm groups,
• So as to maintain a 1 output during A transition

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Dynamic hazards

• Implementation now has a new gate to eliminate
  hazard.

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Dynamic hazards

• In general, all prime implicants must be included
  in the solution to avoid static hazards.
• Quine-McCluskey method best at identifying
  potential static hazards.

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Dynamic hazards

• Exercise
• Obtain a hazard free realisation of the output
  network given by the Karnaugh Map below.
•  

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Summary

• Need to have a Software based approach to
  simplification of Boolean Expressions.
• Need to be aware that the simplified solution may
  not the what is required !