Simplification to Circuit diagram

[Sweps]

Hi Guys.

Iv started this coarse a few days ago and built most of the chips via looking at pre-built diagrams. Iv since cleared them out and am figuring it out by myself.

I am having trouble converting my simplified expression into an actual diagram. For the mux chip I started by getting the SOP from the truth table and then used a karnaugh map to come up with the expression (bc)+(ab)+(a'c).

To me this sais I need 3 AND chips ORed together but from the completed chips I can see it is done with 2 ANDs and an OR.... I cant understand how to convert my expression to a simplified circuit diagram. Been fighting with this all day cant seem to get past this.

Any help would be appreciated.

[Sweps]

Ok so I got a working MUX using that expression but I had to use 3 ANDs a NOT and 3 ORs. Im not sure how people have come up with ways to complete it using fewer chips even after that is the simplest I come up with.

[intelli78]

You need to use Boolean algebra to further simplify the expression. Search for Boolean rules of simplification and apply them to your sum of products.

[Sweps]

Thanks Intelli. I was under the impression that the karnaugh map was used to get it to the smallest expression possible.

So just to clarify, after I have used karnaugh maps to shrink it, I need to use boolean algebra to get it even smaller?

[intelli78]

I am not really familiar with K-Maps, but I imagine they should yield the same result. I think you probably made a mistake with the K-Map somewhere, because the expression can be simplified considerably compared to what you currently have.

[cadet1620]

Sweps wrote

Thanks Intelli. I was under the impression that the karnaugh map was used to get it to the smallest expression possible.

So just to clarify, after I have used karnaugh maps to shrink it, I need to use boolean algebra to get it even smaller?

Karnaugh maps are a easier than Boolean logic.  The key to Karnaugh maps is to find the smallest number of  blocks of "connected" 1s that cover all the 1s.  Connected means 1x2 rectangle, 2x2 square, 1x4 rectangle, etc.

Given the map for Mux:

             | a |
        -----------
        | 0 0 1 1 |
    sel | 0 1 1 0 |
        -----------
           | b |

All of the 1s can be covered with the rectangle on the upper row which is a ~sel and the rectangle on the lower row which is b sel.

Given this map:

             | a |
        -----------
        | 0 0 1 1 |
        | 0 1 1 0 |  d
     c  | 0 1 1 0 |  d
     c  | 0 0 1 0 |
        -----------
           | b |

You need 3 terms, the square in the middle (b d), the vertical rectangle in the third column (a b) and the small rectangle in the top row (a ~c ~d).

Note that it can be tricky to see the connected shapes when they wrap around the edges.  Think about what the term (b ~d) would look like in the above map, or even trickier, (~a ~d).

--Mark

[Sweps]

Ok I must be doing something wrong here.

The SOP i get for the MUX chip is:
('a 'b sel) + (a b 'sel) + ('a b sel) + (a b sel)

My map looks like
     'a'b  'ab   ab   a'b  
'sel               1     1
sel         1     1

I circle the 2 1s at the bottom row and the 2 at the top and the ab-sel going vertical leaving me with:
(b sel) + (a b) + (a 'sel)

This to me sais I need 3 AND gates ORed together which works but I cant get it any smaller?

Note, That map layout I got off a youtube video explaining maps. Is that even the correct way to draw a map?

[cadet1620]

Yes, your k-map is good.  It's the same as this one, just typed instead of graphic.



Circle the 1s in the top row. (red)
Circle the 1s in the bottom row. (blue)
All of the 1s are now in circles; there is no need to do any more circling.
Ignore the grey pair of 1s -- this pair is redundant.
Or the red and blue pairs together to get the minimized result: (a ~sel) + (b sel)

--Mark

[Sweps]

Thanks for all the help guys. That final piece about the redundant 1s was all I was missing!  

[songkeys]

Hello, Sir. I know this is a quite ancient post. But I am really curious about your k-map.
I understand the common style of it, but yours seems kinda unusual and I cant figure it out.
Could you please explain a little more, or just throw me some tutorial links which I haven't found in my searching journey?
Thanks a lot!

[cadet1620]

songkeys wrote

Hello, Sir. I know this is a quite ancient post. But I am really curious about your k-map.
I understand the common style of it, but yours seems kinda unusual and I cant figure it out.
Could you please explain a little more, or just throw me some tutorial links which I haven't found in my searching journey?
Thanks a lot!

That's an older style, the way I learned them back in the '70s, slightly modified for Excel.  The labels on the rows and columns indicate when the named variable is true.  The hand drawn version I learned used braces to group the rows and columns:


--Mark

[songkeys]

Oh, I got it! Thanks for your quick and patient response!!