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Fixing hazards in a logic expression

  1. Mar 28, 2015 #1

    Zondrina

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    1. The problem statement, all variables and given/known data

    Identify and fix all hazards in ##f = (ab + \bar{a}c)(c + \bar{b}) + ab##. Re-check if the final expression obtained is hazard free.

    2. Relevant equations


    3. The attempt at a solution

    So I used a binary tree and found a static 1 hazard present for ##bc = 11##. The hazard was ##a + \bar{a}##.

    To fix this hazard, I need a ##\sum## of ##\Pi## map using ##f##.

    I am confused about the term ##(ab + \bar{a}c)(c + \bar{b})##. I don't know how to place it on a K-Map.

    I know I can't use the distributive laws because that will introduce dynamic hazards since ##b## and ##\bar{b}## are together.

    How would I go about creating the K-Map exactly?
     
  2. jcsd
  3. Mar 28, 2015 #2

    mfb

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    Staff: Mentor

    As you have "+ab" at the end, do you need the "ab" in the first bracket at all?
     
  4. Mar 28, 2015 #3

    Zondrina

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    I would think I do need it. I can't just remove it can I? The first term is an AND term, so I don't have a term of the form ##ab + ab##.
     
  5. Mar 29, 2015 #4

    mfb

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    "((X OR Y) AND Z) OR X" = "(X AND Z) OR (Y AND Z) OR X"
    But "(X AND Z) OR X" is just X.

    Alternatively, just test all 8 options to verify it.
     
  6. Mar 29, 2015 #5

    Zondrina

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    Okay, so I can use distributivity and simplification as long as I'm particularly careful about ##b## and ##\bar{b}##:

    $$(ab + \bar{a}c)(c + \bar{b}) + ab$$
    $$ab(c + \bar{b}) + \bar{a}c(c + \bar{b}) + ab$$
    $$ab(c + \bar{b}) + ab + \bar{a}c(c + \bar{b})$$
    $$ab + \bar{a}c(c + \bar{b})$$

    It's safe to use distributivity now:

    $$ab + \bar{a}cc + \bar{a}c \bar{b}$$
    $$ab + \bar{a}c + \bar{a}c \bar{b}$$
    $$ab + \bar{a}c$$

    Drawing the K-Map:

    ab\c
    0 1
    0 1
    1 1
    0 0

    I see I need a term of the form ##bc## to fill the valley. Hence:

    $$f = ab + \bar{a}c + bc$$

    This expression is hazard free.
     
  7. Mar 29, 2015 #6

    mfb

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    Right.
     
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