LECTURE THIRTEEN PART 1 DIGITAL ELECTRONICS

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LECTURE THIRTEEN PART 1 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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Example of Canonical Conversion

• SOP
•  
• Identify the missing variables in each AND term
• for ?
•  
• for ?
•  
• for ?

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Example of Canonical Conversion

• ? Canonical SOP form
•  
•  
• Two terms the same ? , thus final expression
  is
•  

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Example of Canonical Conversion

• POS
•  
• Identify the missing variables in each term
• for ?
•  
• for ?

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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

• Can simplify expressions like
•  
•  
• by applying rules of Boolean Algebra, POS or SOP
  expression

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Simplification of Boolean Expressions

•     Simplification of switching equation reduces
  the amount of hardware needed to realise the
  function
• ? fewer gates
• ? fewer I.C.s
• ? less cost

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Recap of Karnaugh Map Methods

• A Karnaugh Map is a matrix of squares
• each square represents a minterm or maxterm from
  a Boolean expression
•  
• A Karnaugh Map allows us to find input variable
  redundancies
• ? help reduce output equation

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Recap of Karnaugh Map Methods

• Each map lists the product terms that can be
  formed from n variables, each in a different
  square.
•  
• a product term in n variables is called a minterm
• ? for 3 variables
• ? minterms i.e.
• ? for 4 variables ? minterms.

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Recap of Karnaugh Map Methods

• A map of n variables will have squares,
  each representing a minterm.
• The minterm in each square or element of the map
  is the product of the variables listed at row and
  column of the element
• ? is at the intersection of and

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Recap of Karnaugh Map Methods

• The map is filled by placing 1s in the squares
  for terms that lead to a 1 output.
• ?
• Z1 for (A,B) (0,1) ? put 1 in box
• Z1 for (A,B) (1,0) ? put 1 in box
• put 0 everywhere else.

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Recap of Karnaugh Map Methods

• The Karnaugh Map is very convenient
• provides some feel for the function due to its
  graphical presentation
•  
• However its usefulness is chiefly due to the
  arrangement of the cells or squares.
• Each cell differs from the adjacent cell by
  having exactly one variable complemented in the
  minterm.
• The expression must be written in canonical form
  ? each term in the expression must contain each
  variable.

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Recap of Karnaugh Map Methods

• E.g.
• 3 variables
•  
• The top row BC is numbered such that only 1-bit
  changes at a time, as you move from box to box
• known as a Grey Code (code where only one bit
  changes at a time)
• i.e. not numbered as in the usual sequence, but
  done to show the relationship between variables.

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Recap of Karnaugh Map Methods

• the main reason
• any two squares that are adjacent differ in only
  one variable
• If two adjacent squares both have an entry of one
• ? the corresponding product terms differ in only
  one variable
• ? the two terms can be merged by eliminating that
  variable.

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Recap of Karnaugh Map Methods

• Consider a grouping of two of the circled terms
  which are adjacent we can see that
• ?
• Can represent these 2 terms by , this is
  immediately apparent from the map as for the
  circled ones
• is constant 1
• is constant 0
• changes and ? will not be in the final
  expression.

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Recap of Karnaugh Map Methods

• Then can form final simplified expression from
  the minimum number of circles required to
  encompass all the ones.
•  
•  
• minimum expression
•   we could also include the third circle and thus
  
• but this is not a minimum expression.
• Two loops cover all the ones.

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Recap of Karnaugh Map Methods
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Recap of Karnaugh Map Methods

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Recap of Karnaugh Map Methods

• for which an implementation would be
• One feature should be noted at this point
• The concept of adjacency can be extended to
  include wrapping around the edges of the map.
• ? The maps are considered so that top and bottom
  edges
• and left and right edges are touching.

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Recap of Karnaugh Map Methods

• or we could loop the zeros.

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Recap of Karnaugh Map Methods

• for which an implementation would be

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Recap of Karnaugh Map Methods

• This is equivalent to
• which is equivalent to our first implementation

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NAND and NOR Implementations

• NAND implementation important, due to Silicon
  Area reasons.
•  
• NAND can be made to yield AND and OR gates
• an AND gate can be formed from 2 NAND gates
• and
• ?2-input OR can be formed from 3 NAND gates

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NAND and NOR Implementations

• Thus a set of NANDs can thus be used to make
  any combinational network by substituting the
  above for AND and OR blocks

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NAND and NOR Implementations

• Same applies for NORs

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NAND and NOR Implementations

• Prime implicants and essential prime implicants
  (EPI)

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NAND and NOR Implementations

• Minimum expression
• eqn (1) SOP
• use AND and OR

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NAND and NOR Implementations

• or

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NAND and NOR Implementations

• Sometimes it is easier to group the zeros terms
  together

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NAND and NOR Implementations

• eqn (2)

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NAND and NOR Implementations

• Can further arrange
• eqn (3) POS
• use OR and AND

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NAND and NOR Implementations

• We can also rearrange Z as follows
• eqn (4) Use NOR

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NAND and NOR Implementations

• likewise for sum of products form of Z
• eqn (5) Use NAND

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Implementation of Minimum Functions

• Observing the function Z, from the last example
  again
•  
• To implement Z would require
•     4 inverters
•     11 quad-input AND gates
•     1 eleven-input OR gate
•  
• Thus in general always need to minimise the
  function

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Implementation of Minimum Functions

• Can implement the minimum Z in Sum of Products
  form
• i.e. Eqn. 1 .
•  
• Can implement the minimum Z in negation of Sum of
  Products form
• i.e. Eqn. 2
•  
• Can implement the minimum Z in Product of Sums
  form
• i.e. Eqn. 3

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Implementation of Minimum Functions

• By applying deMorgans Laws we can rearrange Z and
  then implement it
• i.e. Eqn. 4 i.e. inverters NOR gates
•  
• or Egn. 5 i.e. only NAND gates
• Can convert between Product-of-Sums to NOR gate
  implementation, with the use of 2 inverters.
•     between the output of the OR gates and input
  of the AND gate
•     obviously has no effect on the output
•     now each OR NOT ? makes it a NOR
• and
• the NOTs at the input of the AND ?makes it a
  NOR

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Conversion of POS to NOR Gates
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Dont Cares

• When an output value is term is known for every
  possible combination of input variables.
• ? function said to be completely specified.
•  
• However, when output not known for every
  combination of input variables (usually because
  all combinations cannot occur)
• ? function said to be incompletely specified
•  
• The minterms or maxterms that are not used as
  part of the output function are called dont care
  terms.