CO5023 Building Circuits from Truth Tables

Build the following… Let’s say we want a circuit which acts as described by the following truth table: We can express this function easily by looking at all the 1s in the ‘Out’ column. Each 1 can be obtained by ANDing some combination of (A or A’), (B or B’) and (C or C’) It should be understood that A’ means ‘not A’ in this context For example, the first 1 could be obtained with A’.B’.C’ The second is A.B’.C’ Can you complete the rest? To obtain the overall function, we OR all these together So the function A’.B’.C’ + A.B’.C’ + A.B’.C + A.B.C would give us our result For reference X.Y means “X and Y”, X+Y means “X or Y” ABCOut

Make a circuit It’s easy to make this into a circuit as follows: The function was A’.B’.C’ + A.B’.C’ + A.B’.C + A.B.C Test this in logic simulator to check it works

With NAND gates… The same result can be achieved using only NAND gates (which are faster in practice) A NAND of NANDS is equivalent to an OR of ANDs

Sum of products (and product of sums) This technique is know as the sum of products approach to circuit design. It works by ORing a number of ANDs – since mathematicians regard OR as being similar to + and AND as being similar to *, we use the term “sum of products” Sum of products: For each 1 in the output, take the combination of inputs (or their negations) which gives that output when ANDed together Then OR each of the ANDed terms together Remember you can then replace all the OR and AND gates with NANDs in practice. There is also a product of sums approach This works by taking all the zero outputs For each zero output, you should OR the negation of the inputs Then you AND these together Try this for the previous example… (next slide) Use