Since we already have a truth table describing the output of the “good flame” logic circuit, we can simply add another output column to the table to represent the second circuit, and make a table representing the entire logic system: While it is possible to generate a Sum-Of-Products expression for this new truth table column, it would require six terms, of three variables each! Being that there are much fewer instances of a “low” output in the last truth table column, the resulting Product-Of-Sums expression should contain fewer terms. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. To begin, we identify which rows in the last truth table column have “low” (0) outputs, and write a Boolean sum term that would equal 0 for that row’s input conditions. Any other output combination (001, 010, 011, 100, 101, or 110) constitutes a disagreement between sensors, and may therefore serve as an indicator of a potential sensor failure. Thus, they should either all register “low” (000: no flame) or all register “high” (111: good flame). To be able to go from a written specification to an actual circuit using simple, deterministic procedures means that it is possible to automate the design process for a digital circuit. 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). As before, to write down the Boolean expression that describes this truth table, we simply write down the Boolean equation for each line in the truth table where the output is 1. Missed the LibreFest? If using relay circuitry, we could create this AND function by wiring three relay contacts in series, or simply by wiring the three sensor contacts in series, so that the only way electrical power could be sent to open the waste valve is if all three sensors indicate flame: While this design strategy maximizes safety, it makes the system very susceptible to sensor failures of the opposite kind. Implemented in a Programmable Logic Controller (PLC), the entire logic system might resemble something like this: As you can see, both the Sum-Of-Products and Products-Of-Sums standard Boolean forms are powerful tools when applied to truth tables. Suppose that due to the high degree of hazard involved with potentially passing un-neutralized waste out the exhaust of this incinerator, it is decided that the flame detection system be made redundant (multiple sensors), so that failure of a single sensor does not lead to an emission of toxins out the exhaust. Sum-Of-Products, or SOP, Boolean expressions may be generated from truth tables quite easily, by determining which rows of the table have an output of 1, writing one product term for each row, and finally summing all the product terms.This creates a Boolean expression representing the truth table as a whole. 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. Using Boolean algebra techniques, the expression may be significantly simplified: As a result of the simplification, we can now build much simpler logic circuits performing the same function, in either gate or relay form: Either one of these circuits will adequately perform the task of operating the incinerator waste valve based on a flame verification from two out of the three flame sensors. This last sum term represents a 0 output for an input condition of A=1, B=1 and C=1. At minimum, this is what we need to have a safe incinerator system. Each sensor comes equipped with a normally-open contact (open if no flame, closed if flame detected) which we will use to activate the inputs of a logic system: 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. Sum-Of-Products expressions are easy to generate from truth tables. Here, Boolean algebra proves its utility in a most dramatic way. 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. 7.9: Converting Truth Tables into Boolean Expressions, [ "article:topic", "authorname:tkuphaldt", "license:gnu" ], Instructor (Instrumentation and Control Technology). First, though, we must decide what the logical behavior of this control system should be. Whoever is monitoring the incinerator would then exercise judgment in either continuing to operate with a possible failed sensor (inputs: 011, 101, or 110), or shut the incinerator down to be absolutely safe. We can, however, extend the functionality of the system by adding to it logic circuitry designed to detect if any one of the sensors does not agree with the other two. 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. Suppose that due to the high degree of hazard involved with potentially passing un-neutralized waste out the exhaust of this incinerator, it is decided that the flame detection system be made redundant (multiple sensors), so that failure of a single sensor does not lead to an emission of toxins out the exhaust. Thus, our truth table would look like this: It does not require much insight to realize that this functionality could be generated with a three-input AND gate: the output of the circuit will be “high” if and only if input A AND input B AND input C are all “high:”. What we need in this system is a sure way of detecting the presence of a flame, and permitting waste to be injected only if a flame is “proven” by the flame detection system. 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. Suppose we were given the task of designing a flame detection circuit for a toxic waste incinerator. Therefore, the term must be written as (A’ + B’+ C’), because only the sum of the complemented input variables would equal 0 for that condition only: The completed Product-Of-Sums expression, of course, is the multiplicative combination of these two sum terms: Whereas a Sum-Of-Products expression could be implemented in the form of a set of AND gates with their outputs connecting to a single OR gate, a Product-Of-Sums expression can be implemented as a set of OR gates feeding into a single AND gate: Correspondingly, whereas a Sum-Of-Products expression could be implemented as a parallel collection of series-connected relay contacts, a Product-Of-Sums expression can be implemented as a series collection of parallel-connected relay contacts: The previous two circuits represent different versions of the “sensor disagreement” logic circuit only, not the “good flame” detection circuit(s). 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. For a 2-input AND gate, the output Q is true if BOTH input A “AND” input B are both true, giving the Boolean Expression of: ( Q = A and B). The intense heat of the fire is intended to neutralize the toxicity of the waste introduced into the incinerator. Product-Of-Sums expressions lend themselves well to implementation as a set of OR gates (sums) feeding into a single AND gate (product). In other words, a computer could be programmed to design a custom logic circuit from a truth table specification! Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. An example of an SOP expression would be something like this: ABC + BC + DF, the sum of products “ABC,” “BC,” and “DF.”. 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