All we have to do is examine the truth table for any rows where the output is "high" (1), and write a Boolean product term that would equal a value of 1 given those input conditions. Sum-Of-Products expressions are easy to generate from truth tables. An example of an SOP expression would be something like this: ABC + BC + DF, the sum of products "ABC," "BC," and "DF." As you might suspect, a Sum-Of-Products Boolean expression is literally a set of Boolean terms added ( summed) together, each term being a multiplicative ( product) combination of Boolean variables. However, a simple method for designing such a circuit is found in a standard form of Boolean expression called the Sum-Of-Products, or SOP, form. Here, it is not necessarily obvious what kind of logic circuit would satisfy the truth table. The truth table for such a system would look like this: A strategy that would meet both needs would be a "two out of three" sensor logic, whereby the waste valve is opened if at least two out of the three sensors show good flame. It would be nice to have a logic system that allowed for this kind of failure without shutting the system down unnecessarily, yet still provide sensor redundancy so as to maintain safety in the event that any single sensor failed "high" (showing flame at all times, whether or not there was one to detect). That single failure would shut off the waste valve unnecessarily, resulting in lost production time and wasted fuel (feeding a fire that wasn't being used to incinerate waste). Suppose that one of the three sensors were to fail in such a way that it indicated no flame when there really was a good flame in the incinerator's combustion chamber. While this design strategy maximizes safety, it makes the system very susceptible to sensor failures of the opposite kind. Thus, our truth table would look like this: This way, any single, failed sensor falsely showing flame could not keep the valve in the open position rather, it would require all three sensors to be failed in the same manner - a highly improbable scenario - for this dangerous condition to occur. A far better solution would be to design the system so that the valve is commanded to open if and only if all three sensors detect a good flame. If any one of the sensors were to fail in such a way as to falsely indicate the presence of flame when there was none, a logic system based on the principle of "any one out of three sensors showing flame" would give the same output that a single-sensor system would with the same failure. Do we want the valve to be opened if only one out of the three sensors detects flame? Probably not, because this would defeat the purpose of having multiple sensors. First, though, we must decide what the logical behavior of this control system should be. Our task, now, is to design the circuitry of the logic system to open the waste valve if and only if there is good flame proven by the sensors.
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