{"id":25,"date":"2011-12-14T00:26:56","date_gmt":"2011-12-14T05:26:56","guid":{"rendered":"http:\/\/sunapi386.ca\/wordpress\/?p=25"},"modified":"2012-03-29T22:24:22","modified_gmt":"2012-03-30T03:24:22","slug":"reducing-equations","status":"publish","type":"post","link":"https:\/\/sunapi386.ca\/wordpress\/reducing-equations\/","title":{"rendered":"Circuit board that displays the digits of e"},"content":{"rendered":"

Circuit board display first 16 digits of e<\/h1>\n

This is the final project for our CP 120 Digital Electronics lab. We are to build a circuit board display first 16 digits of e. It\u00a0takes in a number (the nth<\/em>\u00a0digit)\u00a0and output that digit of e.<\/em><\/p>\n

In general, we create the logic equation, reduce it with Karnaugh map, program it with a complex programmable logic device (CPLD), and wire it all up to light up a 7-segment\u00a0display.<\/p>\n

Let’s start:<\/p>\n

How does one simplify the work of creating outputs to a 7-segment display?
\nTraditionally, you’d have to use k-maps, and repeat the process for each lighting diode. This means a lot of errors could result.<\/p>\n

So how does one automate the process of creating equations?<\/p>\n

Let’s start of by defining:<\/p>\n

    \n
  1. Inputs<\/li>\n
  2. Outputs<\/li>\n<\/ol>\n


    \nLet’s assume you want to display the first 16 digits of pi – basically one would specify which place you digit you want to call.
    \nExample, the 0th digit would be 3, 1st would be 1, 2nd would be 4, etc.<\/p>\n

    Assuming one uses a binary digit representation of those numbers, 0 -> 0000, 1 -> 0001, 2-> 0010, etc.
    \nYou would map when each diode turns on\/off according to the inputs, let’s call them Q3, Q2, Q1, Q0.<\/p>\n

    \"\"<\/a><\/p>\n

    Since the diode has a, b, c, d, e, f, g as outputs, you’d figure out the equation for each of these.
    \nYou can do so with karnaugh maps, but this is a tedious task.<\/p>\n

    Basically, here is the program that will automate the generation of the equations.<\/p>\n


    \n# Python 3.1<\/code><\/p>\n

    # Let’s define our output, the first 16 digit of pi
    \nPI = [ 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3 ]<\/p>\n

    # Then we need to create a dictionary, of what each number maps to define the inputs one would get for a 7-segment display.
    \ndef segment_7d_dictionary():
    \ndi = {}<\/p>\n

    # We define the nth digit’s relation to the outputs of a to g.
    \n# So digit 0 would have all the diodes on except for diode g
    \n# di[n] = [a, b, c, d, e, f, g]<\/p>\n

    di[0] = [1, 1, 1, 1, 1, 1, 0]
    \ndi[1] = [0, 1, 1, 0, 0, 0, 0]
    \ndi[2] = [1, 1, 0, 1, 1, 0, 1]
    \ndi[3] = [1, 1, 1, 1, 0, 0, 1]
    \ndi[4] = [0, 1, 1, 0, 0, 1, 1]
    \ndi[5] = [1, 0, 1, 1, 0, 1, 1]
    \ndi[6] = [1, 0, 1, 1, 1, 1, 1]
    \ndi[7] = [1, 1, 1, 0, 0, 0, 0]
    \ndi[8] = [1, 1, 1, 1, 1, 1, 1]
    \ndi[9] = [1, 1, 1, 1, 0, 1, 1]<\/p>\n

    return di<\/p>\n

    def cr_7d_display_eqn ( dictionary, constant, node ):
    \nsegmenent_representation = []
    \nfor digit in constant:
    \nsegmenent_representation.append( dictionary[digit][node] )<\/p>\n

    print( segmenent_representation )
    \nm_lines = []
    \nfor i in range( 0, len( segmenent_representation ) ):
    \nif segmenent_representation[i] == 1:
    \nm_lines.append( i )<\/p>\n

    equation = qm.qm( ones = m_lines )
    \nreturn equation<\/p>\n

    Now that we have our functions defined and out of the way, here is the main program:<\/p>\n


    \nimport qm
    \n# To reduce equations http:\/\/robertdick.org\/python\/qm.html<\/code><\/p>\n

    for node in range ( 0, 7 ):
    \ncr_7d_display_eqn ( segment_7d_dictionary(), E, node )<\/p>\n

     <\/p>\n

    So there we have it.
    \nIf you run this, you get:<\/p>\n


    \n[1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1]
    \n['XX0X', 'XXX1']
    \n[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0]
    \n['XX0X', '0XXX', 'XXX0']
    \n[0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1]
    \n['XXX1', '11XX', 'XX1X']
    \n[1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1]
    \n['XX00', 'XX11', 'X10X', '1X0X']
    \n[1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0]
    \n['0X00', '0X11', '100X', 'X101']
    \n[0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1]
    \n['X1X1', '1X1X', '1XX1', '11XX', 'XX11']
    \n[1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1]
    \n['010X', 'XX11', 'XX00', '10XX', '1XX0']<\/code><\/p>\n

     <\/p>\n

    This returns what place of digit it turns on (1), and the corresponding equation.
    \nX = don’t care
    \n0 = negated (let’s define NQ0 as negation of Q0, etc.)
    \n1 = itself
    \nFor the last one
    \n([‘010X’, ‘XX11’, ‘XX00′, ’10XX’, ‘1XX0’])
    \nthe sum of product equation would be:
    \ng = NQ3 BQ2 NQ1 + Q1 Q0 + NQ1 NQ0 + Q3 NQ2 + Q3 NQ0
    \nNow we have simplified the equation with the k-map program, we have to wire it up.<\/p>\n

    After hooking everything up, it looks something like this (note, the PLD is not present):
    \n    \"RC\"<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

    Circuit board display first 16 digits of e This is the final project for our CP 120 Digital Electronics lab. We are to build a circuit board display first 16 digits of e. It\u00a0takes in a number (the nth\u00a0digit)\u00a0and output that digit of e. In general, we create the logic equation, reduce it with Karnaugh … Continue reading Circuit board that displays the digits of e<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[10,9,8],"class_list":["post-25","post","type-post","status-publish","format-standard","hentry","category-hack2600","tag-circuits","tag-k-maps","tag-project"],"_links":{"self":[{"href":"https:\/\/sunapi386.ca\/wordpress\/wp-json\/wp\/v2\/posts\/25","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sunapi386.ca\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sunapi386.ca\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sunapi386.ca\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sunapi386.ca\/wordpress\/wp-json\/wp\/v2\/comments?post=25"}],"version-history":[{"count":14,"href":"https:\/\/sunapi386.ca\/wordpress\/wp-json\/wp\/v2\/posts\/25\/revisions"}],"predecessor-version":[{"id":171,"href":"https:\/\/sunapi386.ca\/wordpress\/wp-json\/wp\/v2\/posts\/25\/revisions\/171"}],"wp:attachment":[{"href":"https:\/\/sunapi386.ca\/wordpress\/wp-json\/wp\/v2\/media?parent=25"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sunapi386.ca\/wordpress\/wp-json\/wp\/v2\/categories?post=25"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sunapi386.ca\/wordpress\/wp-json\/wp\/v2\/tags?post=25"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}