logic gates- sum of products

Discussion in 'Homework Help' started by fan_boy17, Apr 17, 2012.

  1. fan_boy17

    Thread Starter New Member

    Apr 17, 2012
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    cant seem to do this question. Don't understand what the sigma means in this


    2 Find the sum of products expression for the following functions and
    implement using NAND gates.

    F(A,B,C) = Ʃ0,1,3,4,6,7
    F(A,B,C,D) = Ʃ 0,1,2,3,7,8,9,11,12,15
    F(P,Q,R,S) = Ʃ 0,4,5,6,7,8,9,10
     
  2. MrChips

    Moderator

    Oct 2, 2009
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    Sigma means sum, i.e. OR
    The numbers refer to the minterms
    For F(A,B,C), this has three bits. Hence 8 minterms, 0 to 7, or 000 to 111 in binary
    minterm 0 = A' B' C'
    minterm 1 = A' B' C
    etc
     
  3. fan_boy17

    Thread Starter New Member

    Apr 17, 2012
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    i used karnaugh maps to do it and i get an answer of (not B. not c) + (A not. C) + BC

    is that correct as the sum of products expression?
     
  4. MrChips

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    Oct 2, 2009
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    (A . C') is one answer for your middle term.
    (A . B) will also work.
     
  5. WBahn

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    Mar 31, 2012
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    You need to learn how to check your work. Here's how you can do it in this case:

    not B . not C => ABC = x00 = 000 (0) or 100 (4)

    A . not C => ABC = 1x0 = 100 (4) or 110 (6)

    B . C => ABC = x11 = 011 (3) or 111 (7)

    So you have implemented:

    f(A,B,C) = SUM{0, 3, 4, 6, 7}

    It would appear that case 1 is not covered. So let's check that specific case:

    ABC = 1 = 001

    not B . not C => F
    A . not C => F
    B . C => F

    Sure enough, it isn't covered.

    If you start checking your own work every opportunity you can, you will lose a LOT fewer points now and quite possibly avoid killing someone down the road.
     
  6. MrChips

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    Oct 2, 2009
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    WBan is correct. I neglected the 1 term.
    There are two ways to cover this:
    A' . B'
    A' . C
     
  7. WBahn

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    Mar 31, 2012
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    I just realized an inconsistency in the above notation, If the first term, it is written

    not B. something

    and in the second term it is written

    A not. something

    I had interpretted the expression as:

    (B')(C') + (A)(C') + (B)(C)

    But I don't know that this is what fan_boy17 meant. He might have meant


    (B')(C') + (A')(C) + (B)(C)

    Now, this doesn't cover all the terms, either.

    There are two sets of two-input minterms that solve the static logic problem, but you must use all six together if you want to avoid static timing hazards. Depending on the course level you are taking, you may or may not have been introduced the concept of static and dynamic timing hazards and how to deal with them.
     
  8. MrChips

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    Oct 2, 2009
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    Good point.
    I naturally assumed that the OP made a typo and meant to write A . not C = A . C'
    In either case, there will still be a missing term.

    A C' would include minterm6, minterm1 is missing.

    A' C includes minterm1, minterm6 is missing.
     
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