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TECHNICAL ARTICLE B Figure 2 A 0 0 1 0 1 1 1 1 Then, like a truth table, you fill in when the output will be a logical 1 or 0. If we were to write down the Boolean Algebra for this in order to represent each square with a 1 in it, we would get: F = !A.B+A.!B+A.B This is messy because there are four gates and we know we only need one. Using Boolean Algebra would break this down to A+B. So how can the K-Map do the same? The answer is: loops! The idea is to loop together like terms. For example, we can loop together !B.A with B.A. This loop then represents the common term in the loop, ignoring any element that changes. Because B changes this loop simply represents A. B Figure 3 A 0 0 1 0 1 1 1 1 The same can then be done with !A.B and A.B to give a loop that is equal to B. We now have two loops, A and B, so the final solution which we know to be true is: A+B Another example is as follows: B Figure 4 A 0 0 1 1 1 1 0 1 With the loops in place, we can quickly see that the answer is A+!B. What makes K-Maps so useful is that you can quickly see the answers. Now before we move on to bigger K-Maps we need some ground rules to follow! • Loops can only go up or down – NOT diagonally. • Loops contain an EVEN number of terms, however, looping one item is allowed. • - So, only: 1, 2, 4 or 8 terms are allowed per loop! • Loops only wrap common terms. • Loops must always be square or rectangle – no funny blob shapes! • Terms in the loops are AND’ed together. • All Loops are then OR’ed together for the final result. These aren’t the difficult rules you would expect but this is basically all you need to know! So lets jump past three inputs to four inputs as you will hopefully figure out how to do three from this example. My truth table is as follows: Figure 6 EEWeb | Electrical Engineering Community Visit www.eeweb.com 12 D 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 Figure 5 C 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 BA 00 DC 01 00 0 1 01 11 10 1 0 1 1 0 0 A 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F 0 1 0 1 1 0 1 0 0 0 0 0 1 1 1 1 11 10 1 0 0 1 1 1 0 0 TECHNICAL ARTICLE
TECHNICAL ARTICLE To start you should note that the numbers along the top are not in the normal binary sequence. In fact you get a gray code sequence. This is so each row or column only have one term chance from one to the next. So back to our example – the loops in this example give the solution C.D+!A.C+A.!C.!D and from this you can see that the input B has no effect! You can also see in this example how the loops wrap around. Are four inputs enough for your design however? As you can see, K-Maps at this level are only designed to work with four inputs, so to add one or two more means adding a dimension. In effect for a five input system, you will have two K-Maps where one is when the fifth input is logic 0 and the other for when it’s logic 1. However, this can start to become a 3D jigsaw as now it’s possible to loop terms from one level to the next. Next time I’ll be looking at some practical uses of K-Maps and Boolean Algebra in some basic logic circuits. About the Author Paul Clarke is a digital electronics engineer with strong software skills in assembly and C for embedded EEWeb Electrical Engineering Community Join Today www.eeweb.com/register systems. At ebm-papst, he develops embedded electronics for thermal management control solutions for the air movement industry. He is responsible for the entire development cycle, from working with customers on requirement specifications to circuit and PCB design, developing the software, release of drawings, and production support. ■ EEWeb | Electrical Engineering Community Visit www.eeweb.com 13 TECHNICAL ARTICLE
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